a= x+1=4 Q. No. 2 Let T r be the r th term of an A.P., for r = 1,2,3,. If for some positive integers m, n. we 1 1 Option 2 1 1
|
|
|
- Lora Snow
- 5 years ago
- Views:
Transcription
1 Q. No. th term of the sequece, + d, + d,.. is Optio + d Optio + (- ) d Optio + ( + ) d Optio Noe of these Correct Aswer Expltio t =, c.d. = d t = + (h- )d optio (b) Q. No. Let T r be the r th term of A.P., for r =,,,. If for some positive itegers m,. we hve T m = d T =, the T m equls : m Optio m Optio + m Optio Optio 0 Correct Aswer Expltio t =, c.d. = d = + ( m- ) d (I) = + ( - ) d (II) m Solvig (I) d (II) =, d = m m T m = + ( m- ) d = + ( m- ) = m m optio (c) Q. No. If (x +), x d (x + ) re first three terms of AP the its 5 th term is Optio Optio 9 Optio Optio 8 Correct Aswer Expltio x +, x d x + re i A.P. x= x++ x+ x= = x+= d =x- ( x + ) =5 t5 = + d =+ 5= optio (c).
2 Q. No. The sum of first te terms of AP is four times the sum of its first five terms, the rtio of first term d commo differece is Optio Optio Optio Optio Correct Aswer Expltio S 0 =.5s 0 ( +9 b) = 5 ( +d ) +9 b= +8d =8d = d Optio (b) Q. No. 5 The sum of ll odd umbers of two digits is Optio 50 Optio 75 Optio 905 Optio Noe of these Correct Aswer Expltio =, t = 99, d = 99=+ ( -) =5 S 5 ( ) 5 5 = + t = ( +99 ) =75 Optio (b) Q. No. 6 If roots of the equtio x - x + 9x - 8 = 0 re i AP, the its commo differece is Optio ± Optio ± Optio ± Optio ± Correct Aswer Expltio Lets roots re - d,, + d - d+ + + d= (sum of roots) = (- d) ( + d) = d = =7 d =6-7=9 d= ± optio (c)
3 Q. No. 7 If first term of AP is 5, lst term is 5 d the sum of the terms is 00, the the umber of terms is Optio 8 Optio 0 Optio 6 Optio 0 Correct Aswer Expltio Give = 5, t = 5, S =00 S = ( + t ) 00= ( 5+5) =6 optio (c) Q. No. 8 b+ c- c+ - b + b-c If,, re i A.P. the which of the followig is i A.P. b c Optio,b,c Optio,b,c Optio,, b c Optio Noe of these Correct Aswer Expltio b+ c- c+ - b + b-c,, re i A.P. b c b+ c- c+ - b + b-c +, +, + re i A.P. b c + b+ c + b+ c + b+ c,, re i A.P. b c,, re i A.P. b c optio (c) Q. No. 9 If for A.P. T = 8 d T 7 = 0 the S 7 is equl to Optio 6 Optio 6 Optio 06 Optio Noe of these Correct Aswer Expltio 8 = + d.(i) 0 = + 6d.(II) 7 S 7 = ( +6 ) =6
4 Q. No. 0 The first, secod d middle term of AP re, b, c respectively. Sum of ll terms is Optio ( c- ) b- Optio c( c- ) + c b- Optio c( b- ) c- Optio b( c- ) b- Correct Aswer Expltio There is oe middle term. Hece o. of term will be odd. Let o. of term is + t =, d = b -, t + = c + d= c c- = b- + S + = +( b- ) = + + c- ( c-) = + c= c + c b- ( ) ( )( ) ( ) Q. No. If,,.. re i AP where i > 0 i the the vlue of = Optio Optio Optio Optio - + Correct Aswer Expltio Let c.d.= d = d = d = = = d - + -
5 Q. No. The sum of the series -( + d) + ( + d) -( + d) +. upto ( + ) terms is Optio -d Optio + d Optio + d Optio d Correct Aswer Expltio - ( + d ) + ( +d )- ( + d ) ( +d ) st term d term th term = + d + d +... d = + d times Q. No. The fourth term of the sequece,,,. is Optio Optio Optio Optio Noe of these Correct Aswer Expltio,,...reiH.P. Correspodig A.P. is,,, etc th Term of A.P is th Term of H.P is Q. No. Optio + Optio + + Optio The th term of the sequece,,, Optio + Correct Aswer Expltio Optio () stisfies for =,, d. Hece optio () is correct. is
6 Q. No. 5 Sum of first odd turl umbers is Optio + Optio Optio - Optio Noe of these Correct Aswer Expltio =, d = S = ( + ( -) ) = ( + - ) = Optio (b). Q. No. 6 If x, x + d x + re first three terms of G.P., the its th term is Optio 7 Optio -7 Optio 7 - Optio 7 Correct Aswer Expltio (x + ) = x(x + ) (x + ) - x(x + ) = 0 (x + ) (x + - x) = 0 (x + ) -(x + ) = 0 x+ 0 x=- 7 Termsre-,-6,-9,- Optio (c). Q. No. 7 The th term of GP is 8 d the sum of its terms is 55. If is commo rtio is the its first term is Optio Optio Optio 8 Optio Noe of these Correct Aswer Expltio t = 8.r - = 8. - = 8 (I) S = 5 ( - ) r -r =5 - ( ) =55...( II) = Optio (). ( I) ( II)
7 Q. No. 8 If first, secod d eight terms of G.P. re respectively -,, 5, the vlue of is Optio Optio 0 Optio Optio Noe of these Correct Aswer Expltio = -, r= = + - t 8 =r = = 7 +=5 = Optio (c). Q. No. 9 If the sum of ifiitely decresig GP is, d the sum of the squres of its terms is 9, the sum of the cubes of the terms is Optio 05 Optio 08 Optio 79 8 Optio Noe of these Correct Aswer Expltio -r =...( I) 9 =...( II) -r 9 = =... III -r + r + r ( -r ) = - r=+ r + r r=, = = = = -r 6-7 ( )
8 Q. No. 0 If the sum of first two terms of ifiite GP is d every term is twice the sum of ll the successive terms, the its first term is Optio Optio Optio Optio Correct Aswer Expltio Let st term = d commo r term = r r + r= d = - r Solvig these two eq. we will get, r =, = Q. No. Three umbers from icresig GP. If the middle umber is doubled, the the ew umbers re i AP. The commo rtio of the GP is Optio - Optio + Optio - Optio + Correct Aswer Expltio Let three terms re, r, r ( r > ) d, r, r re i A.P. r= + r r - r+=0 ± 6- ± r= = = ± r = ± Q. No. If, b, c, d re i G.P. the + b, b + c, c + d re i Optio A.P. Optio G.P. Optio H.P. Optio Noe of these Correct Aswer Expltio Let commo rtio = r b = r, c =r, d = r + b = ( + r ) b + c = r ( + r ) c + d =.r ( + r ) these re i G.P.
9 Q. No. If the sum of first 6 terms of G.P. is ie times of the sum of its first three terms, the its commo rtio is Optio Optio Optio Optio - Correct Aswer Expltio 6 ( -r ) ( -r ) =9 -r -r ( - r )( + r ) =9( - r ) + r =9 r= Q. No. If p th, q th d r th terms of A.P. re equl to correspodig terms of G.P. d these terms re respectively x, y, z, the x y-z. y z-x, z x-y equls Optio 0 Optio Optio Optio Noe of these Correct Aswer Expltio A.P. st term = c.d = d G.P. st term A. c.r. = R x = + (p-)d = Ar p- y = + (q- )d =AR q- z = + (r- )d = AR r- ( ) ( ) ( ) ( ) x y z = AR AR AR ( ) ( ) q-r d r-p d p-q d y-z z-x x-y p- q- r- ( q r r p p q) d ( )( ) ( )( ) ( )( ) =A R =A R = p q r q r p r p q d
10 Q. No. 5 If, b, c, d re i G.P., the ( + b ) -,(b + c ) -,(c + d ) - re i Optio A.P. Optio G.P. Optio H.P. Optio Noe of these Correct Aswer Expltio Let C.R. = r b = r, c =r d = r ( r ) + b = + ( b ) - + = ( + r ) ( r ) b + c = r + ( b c ) - + = r ( + r ) ( r ) c + d = r 6 + ( c d ) - + = r These re i G.P. ( + r ) 6 Q. No. 6 If, b, c re i G.P. d /x = b /y = c /z the x, y, z re i Optio A.P. Optio G.P. Optio H.P. Optio Noe of these Correct Aswer Expltio, b, c re i G.P. log, log b log c re i A.P. y ( ) Give x = b = cz = k let log = log b= log c=log k x y z b c x=log k, y=log k,z=logk x, y, z re i A.P Q. No. 7 The product of first (odd) terms of G.P. whose middle terms is m is Optio m Optio m Optio m Optio Noe of these Correct Aswer Expltio Let = k + t. t k+ =t. t k = t. t k- =..=(t k+ ) = m
11 ( ) t t... t = m m... k times m= m = m k+ k+ Q. No. 8 Sum of ifiite terms of series is Optio Optio Optio 9 Optio 8 Correct Aswer Expltio S= S= S= = =+ =+ = - S= 9 Q. No. 9 The sum to terms of the series is giveby Optio Optio (+) Optio (+/) Optio Noe of these Correct Aswer Expltio S = S = ( -) S = terms i G.P.
12 + - = = =- S = Optio (). Q. No equls Optio Optio Optio Optio Noe of these Correct Aswer Expltio 99 S = S 00 = S = = =-( ) S = Q. No. If r th term of series is (r + ) -r, the sum of its ifiite terms is Optio 0 Optio 8 Optio 5 Optio 0 Correct Aswer Expltio ( r+) tr= r 5 7 S= S= S= = = = += S=5
13 Q. No. th term of sequece is 8 Optio +( -) Optio - Optio - - Optio Noe of these Correct Aswer Expltio 7 Give sequece is,, It c be writte s,,... 8 =-,-, =-,-,-... t =- Q. No. If AM s re iserted betwee d the rd Optio - Optio Optio - Optio Correct Aswer Expltio Totl o. of terms i correspodig A.P. = 6 =, t 6 = Let c.d. = d = +5 d d = rd AM is th term of A.P. A = + d = + = AM is Q. No. AM s re iserted betwee d 8. If third AM is the is equl to Optio 9 Optio 7 Optio 8 Optio 0 Correct Aswer Expltio =, t + = 8, t =, c.d = d =+ d d= ( ) ( ) 8=+ + d =+ + =6+ =8
14 Q. No. 5 If A, A be two AM s d G, G be two GM s betwee two umbers d b, the A +A is equ lto G G Optio + b b Optio b + b Optio + b b Optio b + b Correct Aswer Expltio, A, A, b re i A.P A + A = + b, G, G, b re i G.P G + G = b A +Α + b = G G b Q. No. 6 If GM s be iserted betwee 60 d 5, the third GM will be Optio 8 Optio 8 Optio 0 Optio 0 Correct Aswer Expltio = 60, t 6 = 5, C.R. = r, G = t =? 5 S=. r r = = 60 r = 60 G = t = r. = =0 8
15 Q. No. 7 If g, g re two G.M.S. betwee two umbers d b, the equl to Optio + b Optio b Optio + b b Optio Noe of these Correct Aswer Expltio, r, r, r G.P. g = r, g = r, b = r r = b r= b g g r r + = + = + r = + b g g r r g g + is equl t ois g g Q. No. 8 If A, A re two A.M.S. betwee two umbers d b, the (A - A )(A - A ) is equl to Optio +b Optio b + b Optio b Optio Noe of these Correct Aswer Expltio, + d, + d, +d A.P. A = + d, A = + d, b= + d. b- d = (A - A )(A - A ) = ( + d - - d) ( + d - - d) = ( + d) = b Q. No. 9 + b The vlue of, for which is A.M. betwee d b is b Optio 0 Optio Optio -/ Optio - Correct Aswer Expltio + b A.M. betwee d b = =
16 Q. No. 0 Six rithmetic mes re iserted betwee d 9/, the th rithmetic me is Optio Optio Optio Optio Correct Aswer Expltio 9 =, t8 =, c. d = d =+7 d d = A = t 5 = + d =+ = Optio (c). Q. No. The sum of ll umbers betwee 00 d 0,000 which re of the form ( N) is equl to Optio 556 Optio 56 Optio 56 Optio Noe of these Correct Aswer Expltio S = S = ( )-( ) 5 = - ( ) ( ) = - 0 =56 Optio (b). Q. No. Sum of term of series is Optio / (+)(+)+ Optio / (+)(+)- Optio / (-)(-)- Optio Noe of these Correct Aswer Expltio t = (- ) ( + ) = - ( +)( +) S = t = ( -) = - 6 = ( )( ) + + = - Optio (b).
17 Q. No. Sum of the series terms, is Optio 560 Optio 570 Optio 580 Optio Noe of these Correct Aswer Expltio S = t- + t S = t + t = t - term =+ - t + - =+ - + ( ( ) ) ( ) = = S 0 = t = = = Q. No. If = 5, the is Optio (5) Optio (5) Optio (5) + 5 Optio Noe of these Correct Aswer Expltio ( ) + =5 ( +) = = 5 Optio (). ( ) Q. No. 5 The sum of series upto 0 terms is Optio Optio Optio Optio oe Correct Aswer Expltio t = ( + ) = ( + + ) = ( ) S = t = = = = = =
18 Q. No. 6 The sum of itegers i betwee d 00 which re divisible by or 5 is Optio 00 Optio 600 Optio 950 Optio 500 Correct Aswer Expltio S divisible by, S divisible by 5, S divisible by 0. required sum=s +S -S = ( +00 ) + ( 5+00) - ( 0+00) =050 Q. No. 7 9 th term of the sequece,,,,5,. is Optio Optio Optio Optio Noe of these Correct Aswer Expltio t 6 = t 5 + t = 8 t 7 = t 6 + t 5 = t 8 = t 7 + t 6 = t 9 = t 8 + t 7 = + = Optio (b) Q. No. 8 The sum of first terms of the series is Optio if is odd d 0 whe is eve Optio - Optio (-) Optio ± Correct Aswer Expltio Sum is whe odd, it is zero whe eve Optio ()
19 Q. No. 9 The sum of first (odd) terms of A.P. whose middle term is m is Optio m Optio m Optio m Optio Noe of these Correct Aswer Expltio Let = K + t + tk+ = t + tk = t + tk- =.. tk+ =m S= t+ t+...+ tk tk+ Kterm = m + m + K times + m = Km + m = m(k + ) = m Optio () Kterm Q. No. 50 The vlue of is Optio Optio 999 Optio 00 Optio oe Correct Aswer Expltio 0.=0... term = term = term = term = = Optio (b)
LEVEL I. ,... if it is known that a 1
LEVEL I Fid the sum of first terms of the AP, if it is kow tht + 5 + 0 + 5 + 0 + = 5 The iterior gles of polygo re i rithmetic progressio The smllest gle is 0 d the commo differece is 5 Fid the umber of
EXERCISE a a a 5. + a 15 NEETIIT.COM
- Dowlod our droid App. Sigle choice Type Questios EXERCISE -. The first term of A.P. of cosecutive iteger is p +. The sum of (p + ) terms of this series c be expressed s () (p + ) () (p + ) (p + ) ()
Lesson-2 PROGRESSIONS AND SERIES
Lesso- PROGRESSIONS AND SERIES Arithmetic Progressio: A sequece of terms is sid to be i rithmetic progressio (A.P) whe the differece betwee y term d its preceedig term is fixed costt. This costt is clled
Objective Mathematics
. If sum of '' terms of a sequece is give by S Tr ( )( ), the 4 5 67 r (d) 4 9 r is equal to : T. Let a, b, c be distict o-zero real umbers such that a, b, c are i harmoic progressio ad a, b, c are i arithmetic
PROGRESSIONS AND SERIES
PROGRESSIONS AND SERIES A sequece is lso clled progressio. We ow study three importt types of sequeces: () The Arithmetic Progressio, () The Geometric Progressio, () The Hrmoic Progressio. Arithmetic Progressio.
VERY SHORT ANSWER TYPE QUESTIONS (1 MARK)
VERY SHORT ANSWER TYPE QUESTIONS ( MARK). If th term of a A.P. is 6 7 the write its 50 th term.. If S = +, the write a. Which term of the sequece,, 0, 7,... is 6? 4. If i a A.P. 7 th term is 9 ad 9 th
07 - SEQUENCES AND SERIES Page 1 ( Answers at he end of all questions ) b, z = n
07 - SEQUENCES AND SERIES Page ( Aswers at he ed of all questios ) ( ) If = a, y = b, z = c, where a, b, c are i A.P. ad = 0 = 0 = 0 l a l
3sin A 1 2sin B. 3π x is a solution. 1. If A and B are acute positive angles satisfying the equation 3sin A 2sin B 1 and 3sin 2A 2sin 2B 0, then A 2B
1. If A ad B are acute positive agles satisfyig the equatio 3si A si B 1 ad 3si A si B 0, the A B (a) (b) (c) (d) 6. 3 si A + si B = 1 3si A 1 si B 3 si A = cosb Also 3 si A si B = 0 si B = 3 si A Now,
Chapter 6. Progressions
Chapter 6 Progressios Evidece is foud that Babyloias some 400 years ago, kew of arithmetic ad geometric progressios. Amog the Idia mathematicias, Aryabhata (470 AD) was the first to give formula for the
CHAPTER - 9 SEQUENCES AND SERIES KEY POINTS A sequece is a fuctio whose domai is the set N of atural umbers. A sequece whose rage is a subset of R is called a real sequece. Geeral A.P. is, a, a + d, a
Mu Alpha Theta National Convention: Denver, 2001 Sequences & Series Topic Test Alpha Division
Mu Alph Thet Ntiol Covetio: Dever, 00 Sequeces & Series Topic Test Alph Divisio. Wht is the commo rtio of the geometric sequece, 7, 9,? 7 (C) 5. The commo differece of the rithmetic sequece,, 0, is 5 (C)
SEQUENCES AND SERIES
9 SEQUENCES AND SERIES INTRODUCTION Sequeces have may importat applicatios i several spheres of huma activities Whe a collectio of objects is arraged i a defiite order such that it has a idetified first
[ 20 ] 1. Inequality exists only between two real numbers (not complex numbers). 2. If a be any real number then one and only one of there hold.
![[ 20 ] 1. Inequality exists only between two real numbers (not complex numbers). 2. If a be any real number then one and only one of there hold.](/thumbs/78/77732161.jpg "[ 20 ] 1. Inequality exists only between two real numbers (not complex numbers). 2. If a be any real number then one and only one of there hold.")
[ 0 ]. Iequlity eists oly betwee two rel umbers (ot comple umbers).. If be y rel umber the oe d oly oe of there hold.. If, b 0 the b 0, b 0.. (i) b if b 0 (ii) (iii) (iv) b if b b if either b or b b if
Q.11 If S be the sum, P the product & R the sum of the reciprocals of a GP, find the value of
Brai Teasures Progressio ad Series By Abhijit kumar Jha EXERCISE I Q If the 0th term of a HP is & st term of the same HP is 0, the fid the 0 th term Q ( ) Show that l (4 36 08 up to terms) = l + l 3 Q3
SUTCLIFFE S NOTES: CALCULUS 2 SWOKOWSKI S CHAPTER 11
UTCLIFFE NOTE: CALCULU WOKOWKI CHAPTER Ifiite eries Coverget or Diverget eries Cosider the sequece If we form the ifiite sum 0, 00, 000, 0 00 000, we hve wht is clled ifiite series We wt to fid the sum
SEQUENCES AND SERIES
Sequeces ad 6 Sequeces Ad SEQUENCES AND SERIES Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives rise to what is called a sequece. Sequeces
Progressions. ILLUSTRATION 1 11, 7, 3, -1, i s an A.P. whose first term is 11 and the common difference 7-11=-4.
Progressios SEQUENCE A sequece is a fuctio whose domai is the set N of atural umbers. REAL SEQUENCE A Sequece whose rage is a subset of R is called a real sequece. I other words, a real sequece is a fuctio
Section 6.3: Geometric Sequences
40 Chpter 6 Sectio 6.: Geometric Sequeces My jobs offer ul cost-of-livig icrese to keep slries cosistet with ifltio. Suppose, for exmple, recet college grdute fids positio s sles mger erig ul slry of $6,000.
Unit 6: Sequences and Series
AMHS Hoors Algebra 2 - Uit 6 Uit 6: Sequeces ad Series 26 Sequeces Defiitio: A sequece is a ordered list of umbers ad is formally defied as a fuctio whose domai is the set of positive itegers. It is commo
Numbers (Part I) -- Solutions
Ley College -- For AMATYC SML Mth Competitio Cochig Sessios v.., [/7/00] sme s /6/009 versio, with presettio improvemets Numbers Prt I) -- Solutios. The equtio b c 008 hs solutio i which, b, c re distict
is an ordered list of numbers. Each number in a sequence is a term of a sequence. n-1 term
Mthemticl Ptters. Arithmetic Sequeces. Arithmetic Series. To idetify mthemticl ptters foud sequece. To use formul to fid the th term of sequece. To defie, idetify, d pply rithmetic sequeces. To defie rithmetic
SEQUENCE AND SERIES NCERT
9. Overview By a sequece, we mea a arragemet of umbers i a defiite order accordig to some rule. We deote the terms of a sequece by a, a,..., etc., the subscript deotes the positio of the term. I view of
Substitute these values into the first equation to get ( z + 6) + ( z + 3) + z = 27. Then solve to get
Problem ) The sum of three umbers is 7. The largest mius the smallest is 6. The secod largest mius the smallest is. What are the three umbers? [Problem submitted by Vi Lee, LCC Professor of Mathematics.
ALGEBRA. Set of Equations. have no solution 1 b1. Dependent system has infinitely many solutions
Qudrtic Equtios ALGEBRA Remider theorem: If f() is divided b( ), the remider is f(). Fctor theorem: If ( ) is fctor of f(), the f() = 0. Ivolutio d Evlutio ( + b) = + b + b ( b) = + b b ( + b) 3 = 3 +
MATH 104 FINAL SOLUTIONS. 1. (2 points each) Mark each of the following as True or False. No justification is required. y n = x 1 + x x n n
MATH 04 FINAL SOLUTIONS. ( poits ech) Mrk ech of the followig s True or Flse. No justifictio is required. ) A ubouded sequece c hve o Cuchy subsequece. Flse b) A ifiite uio of Dedekid cuts is Dedekid cut.
[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms.
![[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms.](/thumbs/82/84777652.jpg "[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms.")
[ 11 ] 1 1.1 Polyomial Fuctios 1 Algebra Ay fuctio f ( x) ax a1x... a1x a0 is a polyomial fuctio if ai ( i 0,1,,,..., ) is a costat which belogs to the set of real umbers ad the idices,, 1,...,1 are atural
S n. = n. Sum of first n terms of an A. P is
PROGREION I his secio we discuss hree impora series amely ) Arihmeic Progressio (A.P), ) Geomeric Progressio (G.P), ad 3) Harmoic Progressio (H.P) Which are very widely used i biological scieces ad humaiies.
SAFE HANDS & IIT-ian's PACE EDT-10 (JEE) SOLUTIONS
. If their mea positios coicide with each other, maimum separatio will be A. Now from phasor diagram, we ca clearly see the phase differece. SAFE HANDS & IIT-ia's PACE ad Aswer : Optio (4) 5. Aswer : Optio
Unit 1. Extending the Number System. 2 Jordan School District
Uit Etedig the Number System Jord School District Uit Cluster (N.RN. & N.RN.): Etedig Properties of Epoets Cluster : Etedig properties of epoets.. Defie rtiol epoets d eted the properties of iteger epoets
EXERCISE - 01 CHECK YOUR GRASP
J-Mathematics XRCIS - 0 CHCK YOUR GRASP SLCT TH CORRCT ALTRNATIV (ONLY ON CORRCT ANSWR). The maximum value of the sum of the A.P. 0, 8, 6,,... is - 68 60 6. Let T r be the r th term of a A.P. for r =,,,...
Mathematical Induction (selected questions)
Mtheticl Iductio (selected questios). () Let P() e the propositio : For P(), L.H.S. R.H.S., P() is true. Assue P() is true for soe turl uer, tht is, () For P( ),, y () By the Priciple of Mtheticl Iductio,
2 n = n=1 a n is convergent and we let. i=1
Lecture 3 : Series So far our defiitio of a sum of umbers applies oly to addig a fiite set of umbers. We ca exted this to a defiitio of a sum of a ifiite set of umbers i much the same way as we exteded
,... are the terms of the sequence. If the domain consists of the first n positive integers only, the sequence is a finite sequence.
Chpter 9 & 0 FITZGERALD MAT 50/5 SECTION 9. Sequece Defiitio A ifiite sequece is fuctio whose domi is the set of positive itegers. The fuctio vlues,,, 4,...,,... re the terms of the sequece. If the domi
ARITHMETIC PROGRESSIONS
CHAPTER 5 ARITHMETIC PROGRESSIONS (A) Mai Cocepts ad Results A arithmetic progressio (AP) is a list of umbers i which each term is obtaied by addig a fixed umber d to the precedig term, except the first
f(bx) dx = f dx = dx l dx f(0) log b x a + l log b a 2ɛ log b a.
Eercise 5 For y < A < B, we hve B A f fb B d = = A B A f d f d For y ɛ >, there re N > δ >, such tht d The for y < A < δ d B > N, we hve ba f d f A bb f d l By ba A A B A bb ba fb d f d = ba < m{, b}δ
Chapter 2 Infinite Series Page 1 of 9
Chpter Ifiite eries Pge of 9 Chpter : Ifiite eries ectio A Itroductio to Ifiite eries By the ed of this sectio you will be ble to uderstd wht is met by covergece d divergece of ifiite series recogise geometric
POWER SERIES R. E. SHOWALTER
POWER SERIES R. E. SHOWALTER. sequeces We deote by lim = tht the limit of the sequece { } is the umber. By this we me tht for y ε > 0 there is iteger N such tht < ε for ll itegers N. This mkes precise
ARITHMETIC PROGRESSION
CHAPTER 5 ARITHMETIC PROGRESSION Poits to Remember :. A sequece is a arragemet of umbers or objects i a defiite order.. A sequece a, a, a 3,..., a,... is called a Arithmetic Progressio (A.P) if there exists
SEQUENCE AND SERIES. Contents. Theory Exercise Exercise Exercise Exercise
SEQUENCE AND SERIES Cotets Topic Page No. Theory 0-0 Exercise - 05-09 Exercise - 0-3 Exercise - 3-7 Exercise - 8-9 Aswer Key 0 - Syllabus Arithmetic, geometric ad harmoic progressios, arithmetic, geometric
11.1 Arithmetic Sequences and Series
11.1 Arithmetic Sequeces ad Series A itroductio 1, 4, 7, 10, 13 9, 1, 7, 15 6., 6.6, 7, 7.4 ππ+, 3, π+ 6 Arithmetic Sequeces ADD To get ext term 35 1 7. 3π + 9, 4, 8, 16, 3 9, 3, 1, 1/ 3 1,1/ 4,1/16,1/
Find a formula for the exponential function whose graph is given , 1 2,16 1, 6
Math 4 Activity (Due by EOC Apr. ) Graph the followig epoetial fuctios by modifyig the graph of f. Fid the rage of each fuctio.. g. g. g 4. g. g 6. g Fid a formula for the epoetial fuctio whose graph is
Name: A2RCC Midterm Review Unit 1: Functions and Relations Know your parent functions!
Nme: ARCC Midterm Review Uit 1: Fuctios d Reltios Kow your pret fuctios! 1. The ccompyig grph shows the mout of rdio-ctivity over time. Defiitio of fuctio. Defiitio of 1-1. Which digrm represets oe-to-oe
1. Hilbert s Grand Hotel. The Hilbert s Grand Hotel has infinite many rooms numbered 1, 2, 3, 4
. Hilbert s Grad Hotel The Hilbert s Grad Hotel has ifiite may rooms umbered,,,.. Situatio. The Hotel is full ad a ew guest arrives. Ca the mager accommodate the ew guest? - Yes, he ca. There is a simple
National Quali cations SPECIMEN ONLY
AH Ntiol Quli ctios SPECIMEN ONLY SQ/AH/0 Mthemtics Dte Not pplicble Durtio hours Totl mrks 00 Attempt ALL questios. You my use clcultor. Full credit will be give oly to solutios which coti pproprite workig.
Student Success Center Elementary Algebra Study Guide for the ACCUPLACER (CPT)
Studet Success Ceter Elemetry Algebr Study Guide for the ACCUPLACER (CPT) The followig smple questios re similr to the formt d cotet of questios o the Accuplcer Elemetry Algebr test. Reviewig these smples
[Q. Booklet Number]
![[Q. Booklet Number]](/thumbs/82/85468362.jpg "[Q. Booklet Number]")
6 [Q. Booklet Numer] KOLKATA WB- B-J J E E - 9 MATHEMATICS QUESTIONS & ANSWERS. If C is the reflecto of A (, ) i -is d B is the reflectio of C i y-is, the AB is As : Hits : A (,); C (, ) ; B (, ) y A (,
1. By using truth tables prove that, for all statements P and Q, the statement
Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3
Chapter 7 Infinite Series
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2
n 2 + 3n + 1 4n = n2 + 3n + 1 n n 2 = n + 1
Ifiite Series Some Tests for Divergece d Covergece Divergece Test: If lim u or if the limit does ot exist, the series diverget. + 3 + 4 + 3 EXAMPLE: Show tht the series diverges. = u = + 3 + 4 + 3 + 3
Solutions to Problem Set 7
8.78 Solutios to Problem Set 7. If the umber is i S, we re doe sice it s relatively rime to everythig. So suose S. Break u the remaiig elemets ito airs {, }, {4, 5},..., {, + }. By the Pigeohole Pricile,
1 Generating functions for balls in boxes
Math 566 Fall 05 Some otes o geeratig fuctios Give a sequece a 0, a, a,..., a,..., a geeratig fuctio some way of represetig the sequece as a fuctio. There are may ways to do this, with the most commo ways
MISCELLANEOUS SEQUENCES & SERIES QUESTIONS
MISCELLANEOUS SEQUENCES & SERIES QUESTIONS Questio (***+) Evaluate the followig sum 30 r ( 2) 4r 78. 3 MP2-V, 75,822,200 Questio 2 (***+) Three umbers, A, B, C i that order, are i geometric progressio
= = =
Sec 5.8 Sec 6. Mathematical Modelig (Arithmetic & Geometric Series) Name: Carl Friedrich Gauss is probably oe of the most oted complete mathematicias i history. As the story goes, he was potetially recogiized
M3P14 EXAMPLE SHEET 1 SOLUTIONS
M3P14 EXAMPLE SHEET 1 SOLUTIONS 1. Show tht for, b, d itegers, we hve (d, db) = d(, b). Sice (, b) divides both d b, d(, b) divides both d d db, d hece divides (d, db). O the other hd, there exist m d
8.3 Sequences & Series: Convergence & Divergence
8.3 Sequeces & Series: Covergece & Divergece A sequece is simply list of thigs geerted by rule More formlly, sequece is fuctio whose domi is the set of positive itegers, or turl umbers,,,3,. The rge of
, 4 is the second term U 2
Balliteer Istitute 995-00 wwwleavigcertsolutioscom Leavig Cert Higher Maths Sequeces ad Series A sequece is a array of elemets seperated by commas E,,7,0,, The elemets are called the terms of the sequece
Infinite Sequences and Series. Sequences. Sequences { } { } A sequence is a list of number in a definite order: a 1, a 2, a 3,, a n, or {a n } or
Mth 0 Clculus II Ifiite Sequeces d Series -- Chpter Ifiite Sequeces d Series Mth 0 Clculus II Ifiite Sequeces d Series: Sequeces -- Chpter. Sequeces Mth 0 Clculus II Ifiite Sequeces d Series: Sequeces
An alternating series is a series where the signs alternate. Generally (but not always) there is a factor of the form ( 1) n + 1
Calculus II - Problem Solvig Drill 20: Alteratig Series, Ratio ad Root Tests Questio No. of 0 Istructios: () Read the problem ad aswer choices carefully (2) Work the problems o paper as eeded (3) Pick
SUTCLIFFE S NOTES: CALCULUS 2 SWOKOWSKI S CHAPTER 11
SUTCLIFFE S NOTES: CALCULUS SWOKOWSKI S CHAPTER Ifiite Series.5 Altertig Series d Absolute Covergece Next, let us cosider series with both positive d egtive terms. The simplest d most useful is ltertig
Mini Lecture 10.1 Radical Expressions and Functions. 81x d. x 4x 4
Mii Lecture 0. Radical Expressios ad Fuctios Learig Objectives:. Evaluate square roots.. Evaluate square root fuctios.. Fid the domai of square root fuctios.. Use models that are square root fuctios. 5.
Topic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or
Topic : Sequeces ad Series A sequece is a ordered list of umbers, e.g.,,, 8, 6, or,,,.... A series is a sum of the terms of a sequece, e.g. + + + 8 + 6 + or... Sigma Notatio b The otatio f ( k) is shorthad
Sets. Sets. Operations on Sets Laws of Algebra of Sets Cardinal Number of a Finite and Infinite Set. Representation of Sets Power Set Venn Diagram
Sets MILESTONE Sets Represetatio of Sets Power Set Ve Diagram Operatios o Sets Laws of lgebra of Sets ardial Number of a Fiite ad Ifiite Set I Mathematical laguage all livig ad o-livig thigs i uiverse
MTH 146 Class 16 Notes
MTH 46 Clss 6 Notes 0.4- Cotiued Motivtio: We ow cosider the rc legth of polr curve. Suppose we wish to fid the legth of polr curve curve i terms of prmetric equtios s: r f where b. We c view the cos si
Putnam Training Exercise Counting, Probability, Pigeonhole Principle (Answers)
Putam Traiig Exercise Coutig, Probability, Pigeohole Pricile (Aswers) November 24th, 2015 1. Fid the umber of iteger o-egative solutios to the followig Diohatie equatio: x 1 + x 2 + x 3 + x 4 + x 5 = 17.
ADDITIONAL MATHEMATICS FORM 5 MODULE 2
PROGRAM DIDIK CEMERLANG AKADEMIK SPM ADDITIONAL MATHEMATICS FORM 5 MODULE 2 PROGRESSIONS (Geometric Progressio) ORGANISED BY: JABATAN PELAJARAN NEGERI PULAU PINANG CHAPTER 2 : GEOMETRIC PROGRESSIONS Cotets
10.5 Test Info. Test may change slightly.
0.5 Test Ifo Test my chge slightly. Short swer (0 questios 6 poits ech) o Must choose your ow test o Tests my oly be used oce o Tests/types you re resposible for: Geometric (kow sum) Telescopig (kow sum)
Lincoln Land Community College Placement and Testing Office
Licol Ld Commuity College Plcemet d Testig Office Elemetry Algebr Study Guide for the ACCUPLACER (CPT) A totl of questios re dmiistered i this test. The first type ivolves opertios with itegers d rtiol
Geometric Sequences. Geometric Sequence. Geometric sequences have a common ratio.
s A geometric sequece is sequece such tht ech successive term is obtied from the previous term by multiplyig by fixed umber clled commo rtio. Exmples, 6, 8,, 6,..., 0, 0, 0, 80,... Geometric sequeces hve
Approximations of Definite Integrals
Approximtios of Defiite Itegrls So fr we hve relied o tiderivtives to evlute res uder curves, work doe by vrible force, volumes of revolutio, etc. More precisely, wheever we hve hd to evlute defiite itegrl
Solutions to Math 347 Practice Problems for the final
Solutios to Math 347 Practice Problems for the fial 1) True or False: a) There exist itegers x,y such that 50x + 76y = 6. True: the gcd of 50 ad 76 is, ad 6 is a multiple of. b) The ifiimum of a set is
Chapter Real Numbers
Chpter. - Rel Numbers Itegers: coutig umbers, zero, d the egtive of the coutig umbers. ex: {,-3, -, -,,,, 3, } Rtiol Numbers: quotiets of two itegers with ozero deomitor; termitig or repetig decimls. ex:
RATIO AND PROPORTION/INDICES/LOGARITHM
Chpter RATIO AND PROPORTION/INDICES/LOGARITHM 006 Novemer. Two umers re i the rtio : d the differece of their squres is 0. The umers re:, 8 6, (c, Noe. If p : is the su-duplicte rtio of p : q, the is:
ANSWERSHEET (TOPIC = ALGEBRA) COLLECTION #2
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC ALGEBRA) COLLECTION # Questio Type A.Sigle Correct Type Q. (B) Sol ( 5 7 ) ( 5 7 9 )!!!! C
PLANCESS RANK ACCELERATOR
PLANCESS RANK ACCELERATOR MATHEMATICS FOR JEE MAIN & ADVANCED Sequeces d Seies 000questios with topic wise execises 000 polems of IIT-JEE & AIEEE exms of lst yes Levels of Execises ctegoized ito JEE Mi
INFINITE SERIES. ,... having infinite number of terms is called infinite sequence and its indicated sum, i.e., a 1
Appedix A.. Itroductio As discussed i the Chpter 9 o Sequeces d Series, sequece,,...,,... hvig ifiite umber of terms is clled ifiite sequece d its idicted sum, i.e., + + +... + +... is clled ifite series
1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2
Mth 3, Clculus II Fil Exm Solutios. (5 poits) Use the limit defiitio of the defiite itegrl d the sum formuls to compute 3 x + x. Check your swer by usig the Fudmetl Theorem of Clculus. Solutio: The limit
{ } { S n } is monotonically decreasing if Sn
Sequece A sequece is fuctio whose domi of defiitio is the set of turl umers. Or it c lso e defied s ordered set. Nottio: A ifiite sequece is deoted s { } S or { S : N } or { S, S, S,...} or simply s {
Intermediate Arithmetic
Git Lerig Guides Iteredite Arithetic Nuer Syste, Surds d Idices Author: Rghu M.D. NUMBER SYSTEM Nuer syste: Nuer systes re clssified s Nturl, Whole, Itegers, Rtiol d Irrtiol uers. The syste hs ee digrticlly
334 MATHS SERIES DSE MATHS PREVIEW VERSION B SAMPLE TEST & FULL SOLUTION
MATHS SERIES DSE MATHS PREVIEW VERSION B SAMPLE TEST & FULL SOLUTION TEST SAMPLE TEST III - P APER Questio Distributio INSTRUCTIONS:. Attempt ALL questios.. Uless otherwise specified, ll worig must be
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION. (2014 Admn. onwards) III Semester. B.Sc. Mathematics CORE COURSE CALCULUS AND ANALYTICAL GEOMETRY
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION (0 Adm. owrds) III Semester B.Sc. Mthemtics CORE COURSE CALCULUS AND ANALYTICAL GEOMETRY Questio Bk & Aswer Key. l l () =... 0.00 b) 0 c). l d =... c
PROBLEM SET 5 SOLUTIONS 126 = , 37 = , 15 = , 7 = 7 1.
Math 7 Sprig 06 PROBLEM SET 5 SOLUTIONS Notatios. Give a real umber x, we will defie sequeces (a k ), (x k ), (p k ), (q k ) as i lecture.. (a) (5 pts) Fid the simple cotiued fractio represetatios of 6
Solutions for May. 3 x + 7 = 4 x x +
Solutios for May 493. Prove that there is a atural umber with the followig characteristics: a) it is a multiple of 007; b) the first four digits i its decimal represetatio are 009; c) the last four digits
SEQUENCES AND SERIES
Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9. Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say that
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
Westchester Community College Elementary Algebra Study Guide for the ACCUPLACER
Westchester Commuity College Elemetry Alger Study Guide for the ACCUPLACER Courtesy of Aims Commuity College The followig smple questios re similr to the formt d cotet of questios o the Accuplcer Elemetry
USA Mathematical Talent Search Round 3 Solutions Year 27 Academic Year
/3/27. Fill i each space of the grid with either a or a so that all sixtee strigs of four cosecutive umbers across ad dow are distict. You do ot eed to prove that your aswer is the oly oe possible; you
Chapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
and each factor on the right is clearly greater than 1. which is a contradiction, so n must be prime.
MATH 324 Summer 200 Elemetary Number Theory Solutios to Assigmet 2 Due: Wedesday July 2, 200 Questio [p 74 #6] Show that o iteger of the form 3 + is a prime, other tha 2 = 3 + Solutio: If 3 + is a prime,
Math 104: Final exam solutions
Mth 14: Fil exm solutios 1. Suppose tht (s ) is icresig sequece with coverget subsequece. Prove tht (s ) is coverget sequece. Aswer: Let the coverget subsequece be (s k ) tht coverges to limit s. The there
MA123, Chapter 9: Computing some integrals (pp )
MA13, Chpter 9: Computig some itegrls (pp. 189-05) Dte: Chpter Gols: Uderstd how to use bsic summtio formuls to evlute more complex sums. Uderstd how to compute its of rtiol fuctios t ifiity. Uderstd how
A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD
Diol Bgoo () A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD I. Itroductio The first seprtio of vribles (see pplictios to Newto s equtios) is ver useful method
DETERMINANT. = 0. The expression a 1. is called a determinant of the second order, and is denoted by : y + c 1
NOD6 (\Dt\04\Kot\J-Advced\SMP\Mths\Uit#0\NG\Prt-\0.Determits\0.Theory.p65. INTRODUCTION : If the equtios x + b 0, x + b 0 re stisfied by the sme vlue of x, the b b 0. The expressio b b is clled determit
THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING
OLLSCOIL NA héireann, CORCAIGH THE NATIONAL UNIVERSITY OF IRELAND, CORK COLÁISTE NA hollscoile, CORCAIGH UNIVERSITY COLLEGE, CORK SUMMER EXAMINATION 2005 FIRST ENGINEERING MATHEMATICS MA008 Clculus d Lier
Test Info. Test may change slightly.
9. 9.6 Test Ifo Test my chge slightly. Short swer (0 questios 6 poits ech) o Must choose your ow test o Tests my oly be used oce o Tests/types you re resposible for: Geometric (kow sum) Telescopig (kow
Data 2: Sequences and Patterns Long-Term Memory Review Grade 8 Review 1
Review 1 Word Bak: use these words to fill i the blaks for Questio 1. Words may be used oce, more tha oce, or ot at all. sequece series factor term equatio expressio 1. A() is a set of umbers or objects
Exam 2 CMSC 203 Fall 2009 Name SOLUTION KEY Show All Work! 1. (16 points) Circle T if the corresponding statement is True or F if it is False.
1 (1 poits) Circle T if the correspodig statemet is True or F if it is False T F For ay positive iteger,, GCD(, 1) = 1 T F Every positive iteger is either prime or composite T F If a b mod p, the (a/p)
= 4 and 4 is the principal cube root of 64.
Chapter Real Numbers ad Radicals Day 1: Roots ad Radicals A2TH SWBAT evaluate radicals of ay idex Do Now: Simplify the followig: a) 2 2 b) (-2) 2 c) -2 2 d) 8 2 e) (-8) 2 f) 5 g)(-5) Based o parts a ad
UNIT #5 SEQUENCES AND SERIES COMMON CORE ALGEBRA II
Awer Key Nme: Dte: UNIT # SEQUENCES AND SERIES COMMON CORE ALGEBRA II Prt I Quetio. For equece defied by f? () () 08 6 6 f d f f, which of the followig i the vlue of f f f f f f 0 6 6 08 (). I the viul
Things I Should Know In Calculus Class
Thigs I Should Kow I Clculus Clss Qudrtic Formul = 4 ± c Pythgore Idetities si cos t sec cot csc + = + = + = Agle sum d differece formuls ( ) ( ) si ± y = si cos y± cos si y cos ± y = cos cos ym si si
Math 2412 Review 3(answers) kt
Math 4 Review 3(aswers) kt A t A e. If the half-life of radium is 690 years, ad you have 0 grams ow, how much will be preset i 50 years (rouded to three decimal places)?. The decay of radium is modeled