SEQUENCE AND SERIES. Contents. Theory Exercise Exercise Exercise Exercise

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1 SEQUENCE AND SERIES Cotets Topic Page No. Theory 0-0 Exercise Exercise Exercise Exercise Aswer Key 0 - Syllabus Arithmetic, geometric ad harmoic progressios, arithmetic, geometric ad harmoic meas, sums of fiite arithmetic ad geometric progressios, ifiite geometric series, sum of squares ad cubes of the first atural umbers. Name : Cotact No. ARRIDE LEARNING ONLINE E-LEARNING ACADEMY A-79 idra Vihar, Kota Rajastha 3005 Cotact No

2 DEFINITION : SEQUENCE AND PROGRESSION KEY CONCEPTS A sequece is a set of terms i a defiite order with a rule for obtaiig the terms. e.g., /, /3,..., /,... is a sequece. AN ARITHMETIC PROGRESSION (AP) : AP is sequece whose terms icrease or decrease by a fixed umber. This fixed umber is called the commo differece. If a is the first term & d the commo differece, the AP ca be writte as a, a + d, a + d,... a + (-) d,... th term of this AP t = a + (-) d, where d = a a - The sum of the first terms of the AP is give by ; S = [a + ( - )d] = [a +l] Where l is the last term. Notes: (i) If each term of a A.P. is icreased, decreased, multiplied or divided by the same ozero umber, the the resultig sequece is also a AP. (ii) Three umbers i AP ca be take as a - d, a, a + d ; four umbers i AP ca be take as a, a - 3d, a - d, a + d, a + 3d five umbers i AP are a - d, a - d, a, a +d, a +d & six terms i AP are a - 5d, a - 3d, a - d, a + d, a + 3d, a + 5d etc. (iii) The commo differece ca be zero, positive or egative. (iv) The sum of the two terms of a AP equidistat from the begiig & ed is costat ad equal to the sum of first & last terms. (v) Ay term of a AP (except the first ) is equal to half the sum of terms which are equidistat from it. a = / (a- k + a+ k), k < For k =, a = (/ ) (a- + a+ ) ; For =, a = (/ ) (a a ) ad so o. k (vi) t r = S r - S r- (vii) If a, b, c are i AP Þ b = a + c. GEOMETRIC PROGRESSION (GP) : GP is a sequece of umbers whose first term is o-zero & each of the succeedig terms is equal to to the proceedig terms multiplied by a costat. Thus i a GP the ratio of successive term is costat. This costat factor is called the COMMON RATIO of the series & is obtaied by dividig ay term by that which immediately proceeds it. Therefore a, ar, ar, ar 3, ar,... is a GP with a as the first term & r as commo ratio. (i) th term = a r ( - ) (ii) Sum of the I st a r terms i.e. S =, if r ¹, r - (iii) sum of ifiite GP whe r < whe r 0 if r < therefore, S = a r < - r (iv) If each term of a GP be multiplied or divided by the same o-zero, quatity, the resultig sequece is also a GP. (v) Ay 3 cossecutive terms of a GP ca be take as a/r, a, ar ; Ay cossecutive Arride learig Olie E-learig Academy A-79 Idra Vihar, Kota Rajastha 3005 Page No. #

3 (v) Ay 3 cossecutive terms of a GP ca be take as a/r, a, ar ; Ay cossecutive terms of a GP ca be take as a/r 3, a/r, ar, ar 3 & so o. = (vi) If a, b, c are i GP Þ b ac. HARMONIC PROGRESSION (HP) : A sequece is said to HP if the reciprocals of its terms are i AP. If the sequece a, a, a 3,..., a is a HP the /a, /a,..., /a is a AP & coverse. Here we do ot have the formula for the sum of the terms of a HP. For HP whose first term is a & secod term is b, the th term is t ab =. b + ( - )(a - b) If a, b, c are i HP Þ b = ac a+ c or a a-b = c b-c AIRTHMATIC MEAN : MEANS If three terms are i AP the the middle term is called the AM betwee the order two, so if a, b, c are i AP, b is AM of a & c. a + a + a a AM for ay positive umber a, a,...a is ; A = = 3 - AIRTHMATIC MEANS BETWEEN TWO NUMBERS : If a,b are ay two give umbers & a, A, A,..., A, b are i AP the A, A,...A are the AM s betwee a & b. A = a + b - a + A, = a+ (b -a) +,..., A = a+ (b -a) + b-a = a + d, = a + d,..., A = a + d, where d = + Note : Sum of AM s iserted betwee a & b is equal to times the sigle AM betwee a & b i.e. å Ar = A where A is the sigle AM betwee a & b. r= GEOMETRIC MEANS : If a, b, c are i GP, b is the GM betwee a & c. b = ac, therefore b - GEOMETRIC MEANS BETWEEN a, b : = ac ; a > 0, c > 0. If a, b are two give umbers & a, G, G,..., G, b are i GP. The G, G, G 3,...G are GMs betwee a & b. G =, G a(b/ a / + =,..., G a(b/ a / + = / + a(b / a) ) ) = ar, = ar,... = ar, where r = (b/a) /+ Note : The product of GMs betwee a & b. is equal to th power of the sigle GM betwee a & b i.e. p r= G = (G) r where G is the sigle GM betwee a & b Arride learig Olie E-learig Academy A-79 Idra Vihar, Kota Rajastha 3005 Page No. #

4 HARMONIC MEAN : If a, b, c are i HP, betwee a & c, the b = ac / [a + c]. Relatio betwee meas : (i) If A, G, H are respectively A.M., G.M., H.M. betwee a & b both beig uequal & positive the, G² = AH (i.e. A, G, H are i G.P.) ad A ³ G ³ H. (ii) A.M. ³ G.M. ³ H.M. Let a, a, a 3,...a be positive real umbers, the we defie their a + a + a a A.M. =, their G.M. = (a a a 3...a ) / ad their H.M. = a + a a. It ca be show that A.M. ³ G.M. ³ H.M. ad equality holds at either places iff a = a = a 3 =...= a ARITHMETICO - GEOMETRIC SERIES : A series each term of which is formed by multiplyig the correspodig term of a AP & GP is called the Arithmetico-Geometric Series, e.g. + 3x + 5x + 7x Here, 3, 5,... are i AP &, x, x, x 3... are i GP.. Sum of terms of a Arithmetico-Geometric Series : Let the S = a + (a + d)r + (a + d)r - a dr ( - r ) [a + ( -)d] r [a + ( -)d] r S = + -, r ¹ -r (-r) -r - SUM TO INFINITY : If r < & the Lim r a dr = 0. S = + -r (-r) SIGMA NOTATIONS THEOREM : (i) å (a ± b ) = å a ± å b (ii) r r r r r= r= r= å k ar = k å ar (iii) r= r= r= å k = k ; where k is a costat Arride learig Olie E-learig Academy A-79 Idra Vihar, Kota Rajastha 3005 Page No. # 3

5 RESULTS ( + ) (i) å r = (sum of the first atural os.) r= ( + )( + ) (ii) å r = (sum of the squares of the first atural umbers) r= 6 3 ( + ) é ù (iii) r = r r= ê ú ër= û (iv) å å ( sum of the cubes of the first atural umbers) å r= 30 r = ( + )( + )( ) Method of differece for fidig th term : Let u, u, u 3... be a sequece, such that u u, u 3 u,... is either a A.P. or a G.P. the th term u of this sequece is obtaied as follows S = u + u + u u...(i) S = u + u u + u...(ii) (i) (ii) Þ u = u + (u u ) + (u 3 u ) (u u ) Where the series (u u ) + (u 3 u ) (u u ) is either i A.P. or i G.P. the we ca fid u. Note : The above method ca be geeralised as follows : Let u, u, u 3,... be a give sequece. The first differes aee D u, D u, D u 3,... where D u = u u, D u = u u etc. The secod differeces are D u, D u, D u 3,..., where D u = D u D u, D u = D u 3 D u etc. This process is cotiued uitil the k th differeces D k u, D k u,... are obtaied, where the k th differece are all equal or they form a GP with commo ratio differet form. Case - : The k th differeces are all equal. I this case the th term, u is give by u = a 0 k + a k a k, where a 0, a,..., a k are calculated by usig first 'k + ' terms of the sequece. Case - : The k th differeces are i GP with commo ratio r (r ¹ r) The th term is give by u = l r + a 0 k + a k a k Method of differece for fidig s : If possible express r th term as differece of two terms as t r = ± (f(r) f(r ± )). This ca be explaied with the help of examples give below. Þ t = f() f(0), t = f() f(),... t = f() f(-) S = f() f(0) Arride learig Olie E-learig Academy A-79 Idra Vihar, Kota Rajastha 3005 Page No. #

6 PART - I : OBJECTIVE QUESTIONS * Marked Questios are havig more tha oe correct optio. Sectio (A) : Arithmetic Progressio A-. Which term of the series is 98- (A) 95 th (B) 00 th (C) 0 th (D) 0 th A-. A-3. A-. A-5. If fourth term of a A.P. is thrice its first term ad seveth term (third term) =, the its commo differece is- (A) (B) (C) (D) 3 The first term of a A.P. of cosecutive iteger is p +. The sum of (p + ) terms of this series ca be expressed as (A) (p + ) (B) (p + ) (p + ) (C) (p + ) 3 (D) p 3 + (p + ) 3 The sum of itegers from to 00 that are divisible by or 5 is (A) 550 (B) 050 (C) 3050 (D) oe of these If a, a, a 3,... are i A.P. such that a + a 5 + a 0 + a 5 + a 0 + a = 5, the a + a + a a 3 + a is equal to (A) 909 (B) 75 (C) 750 (D) 900 A-6. The iterior agles of a polygo are i A.P. If the smallest agle is 0º & the commo differece is 5º, the the umber of sides i the polygo is: (A) 7 (B) 9 (C) 6 (D) oe A-7. A-8. Cosider a A.P. with first term 'a' ad the commo differece 'd'. Let S k deote the sum of its first K Skx terms. If S is idepedet of x, the x (A) a = d/ (B) a = d (C) a = d (D) oe There are A.M's betwee 3 ad 5, such that the 8th mea: ( - ) th mea:: 3: 5. The value of is. (A) (B) 6 (C) 8 (D) 0 A-9. The A.M. betwee two umbers is A, ad S is the sum of arithmetic meas betwee these umbers, the : (A) S = A (B) A = S (C) A = S (D) oe of these A-0. If the root of the equatio x 3 x + 39 x 8 = 0, are i A.P., the their commo differece, will be : (A) ± (B) ± (C) ± 3 (D) ± A-*. For the A.P. give by a, a,..., a,..., the equatios satisfied are (A) a + a + a 3 = 0 (B) a - a + a 3 = 0 (C) a + 3a - 3a 3 - a = 0 (D) a - a + 6a 3 - a + a 5 = 0 Arride learig Olie E-learig Academy A-79 Idra Vihar, Kota Rajastha 3005 Page No. # 5

7 Sectio (B): Geometric progeressio B-. The third term of a G.P is. The product of the first five terms is (A) 3 (B) 5 (C) (D) oe of these B-. The sum of the series log + log + log log is (A) ( + ) (B) ( + ) ( + ) (C) ( + ) (D) ( + ) B-3. a, b be the roots of the equatio x 3x + a = 0 ad g, d the roots of x x + b = 0 ad umbers a, b, g, d (i this order) form a icreasig G.P., the (A) a = 3, b = (B) a =, b = 3 (C) a =, b = 3 (D) a =, b = 6 B-. The ratioal umber, which equals the umber. 357 with recurrig decimal is 355 (A) 00 (B) (C) 999 (D) oe of these 9 B-5*. If sum of the ifiite G.P., p,,, p, 3,... is, the value of p is p p (A) 3 (B) 3 (C) 3 (D) 3 B-6*. Idicate the correct alterative(s), for 0 < f < p/, if: x = å = 0 cos f, y = å = 0 si f, z = å = 0 cos f si f the: (A) xyz = xz + y (B) xyz = xy + z (C) xyz = x + y + z (D) xyz = yz + x Sectio (C) : Harmoic Progressio, AGP, Relatio betwee AM, GM, HM & Miscellaeous C-. Suppose a, b, c are i A.P. & ½a½, ½b½, ½c½ <. If x = + a + a +... to ; y = + b + b +... to & z = + c + c +... to the x, y, z are i: (A) A.P. (B) G.P. (C) H.P. (D) oe C-*. If positive umbers a, b, c are i A.P. ad a, b, c are i H.P., the (A) a = b = c (B) b = a + c (C) b = ac 8 (D) oe of these C-3. If the sum of the roots of the quadratic equatio, ax + bx + c = 0 is equal to sum of the squares of their reciprocals, the a c, b a, c b are i (A) A.P. (B) G.P. (C) H.P. (D) oe Arride learig Olie E-learig Academy A-79 Idra Vihar, Kota Rajastha 3005 Page No. # 6

8 C-. If a(b - c)x + b(c -a)x + c(a -b) = 0 has equal roots, the a,b,c are i : (A) A.P (B) G.P (C) H.P (D) oe of these C-5. If a x = b y = c z = d t ad a, b, c, d are i G.P., the x, y, z, t are i (A) A.P. (B) G.P. (C) H.P. (D) oe of these C-6. If 3 + (3 + d) + (3 + d) upto = 8, the the value of d is: (A) 9 (B) 5 (C) (D) oe of these C-7. The H.M. betwee two umbers is, their A.M. is A ad G.M. is G. If A + G = 7, the the umbers are : (A) 8, (B) 8, 6 (C) 6, 3 (D) 6, C-8. If A, G & H are respectively the A.M., G.M. & H.M. of three positive umbers a, b, & c, the the equatio whose roots are a, b, & c is give by: (A) x 3-3 Ax + 3 G 3 x - G 3 = 0 (B) x 3-3 Ax + 3 (G 3 /H)x - G 3 = 0 (C) x Ax + 3 (G 3 /H) x - G 3 = 0 (D) x 3-3 Ax - 3 (G 3 /H) x + G 3 = 0 C-9*. If the arithmetic mea of two positive umbers a & b (a > b) is twice their geometric mea, the a: b is: (A) + 3 : - 3 (B) : (C) : 7-3 (D) : 3 C-0. The sum å is equal to: r = r - (A) (B) 3/ (C) /3 (D) oe C-*. If år (r + ) (r + 3) = a + b 3 + c + d + e, the r= (A) a + c = b + d (B) e = 0 (C) a, b /3, c are i A.P. (D) c/a is a iteger C-. The sum of the first -terms of the series is ( + ), whe is eve. Whe is odd, the sum is (A) ( + ) (B) ( + ) (C) ( + ) (D) ( + ) Arride learig Olie E-learig Academy A-79 Idra Vihar, Kota Rajastha 3005 Page No. # 7

9 PART - II : MISCELLANEOUS OBJECTIVE QUESTIONS Comprehesio # We kow that = ( + ) = f(), = ( + )( + ) 6 = g(), = æ ( + ) ö ç è ø = h(). g() g( ) must be equal to (A) (B) ( ) (C) (D) 3. Greatest eve atural umber which divides g() f(), for every ³, is (A) (B) (C) 6 (D) oe of these 3. f() + 3 g() + h() is divisible by (A) oly if = (B) oly if is odd (C) oly if is eve (D) for all Î N Comprehesio # There are + terms i a sequece of which first + are i A. P. ad last + are i G. P. the commo differece of A. P. is ad commo ratio of G. P. is /. The middle term of the A. P. is equal to middle term of G. P. Let middle term of the sequece is a m ad a m is the sum of ifiite G. P. Whose sum of first two terms is F 5I HG K J ad ratio of these terms is First term of give ifiite G. P. is equal to : (A) / (B) (C) (D) 9/6 5. Number of terms i the give sequece is equal to : (A) 9 (B) 7 (C) 3 (D) oe of these 6. Middle term of the give sequece, i.e. a m is equal to : (A) 6/7 (B) 3/7 (C) 8/7 (D) 6/9 7. First term of give sequece is equal to : (A) 8/7, 0/7 (B) 36/7 (C) 36/7 (D) 8/7 8. Middle term of give A. P. is equal to : (A) 6/7 (B) 0/7 (C) 78/7 (D) Arride learig Olie E-learig Academy A-79 Idra Vihar, Kota Rajastha 3005 Page No. # 8

10 Match the colum 9. Colum I Colum II (A) If log 5, log 5 ( x 5) ad log 5 ( x 7/) are i A.P., (p) 6 the value of x is equal to (B) Let S deote sum of first terms of a A.P. If S = 3S, (q) 9 S3 the S is 8 6 (C) Sum of ifiite series is (r) 3 3 (D) The value of = (s) 0. Colum - I Colum - II (A) If a i 's are i A.P. ad a + a 3 + a + a 5 + a 7 = 0, a is equal to (p) (B) Sum of a ifiite G.P. is 6 ad it's first term is 3. the harmoic mea of first ad third terms of G.P. is (q) (C) å r rr is equal to ( - ) = (r) / (D) If roots of the equatio x 3 ax + bx + 7= 0, are i G.P. with commo ratio, the a + b is (s) 6/5 ASSERSION-REASON TYPE This sectio cotais questios umbered QNo.(-3) to Each questio cotais Statemet (Assertio) ad Statemet (Reaso). For the followig questios aswers (A), (B), (C) ad (D) are give below, of which oly oe is correct. (A) Statemet - is true, Statemet - is true ; Statemet - is correct explaatio for Statemet - (B) Statemet - is true, Statemet - is true ; Statemet - is NOT correct explaatio for Statemet - (C) Statemet - is true, Statemet - is false. (D) Statemet - is false, Statemet - is true.. STATEMENT- :,,, 8,... is a G.P.,, 8, 6, 3 is a G.P. ad +, + 8, + 6, 8 + 3,... is also a G.P. STATEMENT- : Let geeral term of a G.P. with commo ratio r be T k + ad geeral term of aother G.P. with commo ratio r be T k +, the the series whose geeral term T k + = T k + + T k + is also a G.P. with commo ratio r.. Statemet : 3,6, are i G.P., the 9,,8 are i H.P. Statemet : If middle term is added i three cosecutive terms of a G.P, resultat will be i H.P. 3. Statemet - : I the set of atural umbers sum of first '' prime umbers is eve or odd accordig as is odd or eve respectively. Statemet - : Sice all prime umbers are odd, sum is eve whe umber of primes are eve ad odd whe umber of primes are odd. Arride learig Olie E-learig Academy A-79 Idra Vihar, Kota Rajastha 3005 Page No. # 9

11 PART - I : OBJECTIVE QUESTIONS. The sum of terms of a A.P. is a( ). The sum of the squares of these terms is : (A) a ( ) (B) a ( - ) ( - ) 6 (C) a ( ) ( ) (D) a ( + ) ( + ) 3. If a, a, a 3,... are the terms of a AP such that a a 7 a6 = the 3 a8 is : (A) 3 (B) If a, a,... a are i A.P. with commo differece d ¹ 0, the the sum of the series (si d) [cosec a cosec a + cosec a cosec a cosec a cosec a ] (C) 5 (D) (A) sec a sec a (C) cot a cot a (B) cosec a cosec a (D) ta a ta a. I a G.P. of positive terms, ay term is equal to the sum of the ext two terms. The commo ratio of the G.P. is (A) cos 8 (B) si 8 (C) cos 8 (D) si 8 5. If a, a, a, a 3,..., a, b are i A.P. ad a, g, g, g 3,...g, b are i G.P. ad h is the harmoic mea of a ad b, the a + a g g + a + a g g a + a g g + + is equal to (A) h (B) h (C) h (D) h 6. Oe side of a equilateral triagle is cm. The mid-poits of its sides are joied to form aother triagle whose mid - poits are i tur joied to form still aother triagle. This process cotiues idefiitely. The the sum of the perimeters of all the triagles is (A) cm (B) cm (C) 88 cm (D) oe of these 7. If the sum of terms of a G.P. (with commo ratio r) begiig with the p th term is k times the sum of a equal umber of terms of the same series begiig with the q th term, the the value of k is: (A) r p/q (B) r q/p (C) r p - q (D) r p + q 8. If A, A be two A.M.s ad G, G be two G.M.s betwee a ad b, the A + A G G is equal to (A) a + b ab (B) ab a + b (C) a + b ab (D) a + b ab 9. If P, Q be the A.M., G.M. respectively betwee ay two ratioal umbers a ad b, the P Q is equal to (A) a -b a (B) a + b (C) ab a + b æ a (D) ç - è b ö ø Arride learig Olie E-learig Academy A-79 Idra Vihar, Kota Rajastha 3005 Page No. # 0

12 0. If x i > 0, i =,,..., 50 ad x + x x 50 = 50, the the miimum value of x + x x equal to 50 (A) 50 (B) (50) (C) (50) 3 (D) (50). If there are H.M. betwee ad ad the ratio of 7 th H.M. to ( ) th H.M. is 9 : 5, the will be : 3 (A) (B) 3 (C) (D) 5. If + + p +... upto = 3 6, the = (A) p / (B) p / (C) p /8 (D) oe of these 3. Sum of the series S = is (A) (B) (C) (D) oe of these If H = , the value of is (A) H (B) + H (C) H (D) H + S 3(+ 8S) 5. If S, S, S 3 are the sums of first atural umbers, their squares, their cubes respectively, the S is equal to (A) (B) 3 (C) 9 (D) 0 Multiple choice 6. The sides of a right triagle form a G.P. The taget of the smallest agle is (A) 5 + (B) 5 - (C) 5 + (D) 5-7. If b, b, b 3 (b i > 0) are three successive terms of a G.P. with commo ratio r, the value of r for which the iequality b 3 > b 3b holds is give by (A) r > 3 (B) 0 < r < (C) r = 3.5 (D) r = 5. PART - II : SUBJECTIVE QUESTIONS. I a A.P. the third term is four times the first term, ad the sixth term is 7 ; fid the series.. The third term of a A.P. is 8, ad the seveth term is 30 ; fid the sum of 7 terms. 3. How may terms of the series 9, 6, 3,... must be take that the sum may be 66?. Fid the umber of itegers betwee 00 & 000 that are (i) divisible by 7 (ii) ot divisible by 7 5. Fid the sum of all those itegers betwee 00 ad 800 each of which o divisio by 6 leaves the remaider Fid the sum of 35 terms of the series whose p th term is 7 p The sum of three umbers i A.P. is 7, ad their product is 50, fid them. Arride learig Olie E-learig Academy A-79 Idra Vihar, Kota Rajastha 3005 Page No. #

13 8. If a, b, c are i A.P., the show that: (i) a (b + c), b (c + a), c (a + b) are also i A.P. (ii) b + c - a, c + a - b, a + b - c are i A.P. 9. Show that, 3, 5 caot be the terms of a sigle A.P. 0. If the sum of m terms of a A.P. is equal to the sum of the terms ad the ext p terms, the prove that æ ö (m + ) ç - è m p ø æ ö = (m + p) ç -. è m ø. The third term of a G.P. is the square of the first term. If the secod term is 8, fid its sixth term.. The cotiued product of three umbers i G.P. is 6, ad the sum of the products of them i pairs is 56; fid the umbers 3. If the p th, q th, r th terms of a G.P. be a, b, c respectively, prove that a q r b r p c p q =.. The sum of three umbers which are cosecutive terms of a A.P. is. If the secod umber is reduced by & the third is icreased by, we obtai three cosecutive terms of a G.P., fid the umbers. 5. If the p th, q th & r th terms of a AP are i GP. Fid the commo ratio of the GP. 6. The sum of ifiite umber of terms of a G.P. is ad the sum of their cubes is 9. Fid the series. 7. If a, b, c, d are i G.P., prove that : (i) (ii) (a - b ), (b - c ), (c - d ) are i G.P.,, are i G.P. a + b b + c c + d 8. I a circle of radius R a square is iscribed, the a circle is iscribed i the square, a ew square i the circle ad so o for times. Fid the limit of the sum of areas of all the circles ad the limit of the sum of areas of all the squares as. 9. The sum of the first te terms of a AP is 55 & the sum of first two terms of a GP is 9. The first term of the AP is equal to the commo ratio of the GP & the first term of the GP is equal to the commo differece of the AP. Fid the two progressios. 0. If 0 < x < p ad the expressio exp {( + ½cos x½ + cos x + ½cos 3 x½ + cos x +... upto ) log e } satisfies the quadratic equatio y 0y + 6 = 0 the fid the value of x.. Fid the th term of a H.P. whose 7 th term is 0 ad 3 th term is 38.. Give that a, g are roots of the equatio, A x - x + = 0 ad b, d the roots of the equatio, B x - 6 x + = 0, fid values of A ad B, such that a, b, g & d are i H.P. 3. Sum the followig series 3 (i) to terms (ii) to ifiity Fid the sum of terms of the series the r th term of which is (r + ) r Fid the sum of the series + 3 (3) (3) 3 + (3) +... up to Arride learig Olie E-learig Academy A-79 Idra Vihar, Kota Rajastha 3005 Page No. #

14 6. The arithmetic mea of two umbers is 6 ad their geometric mea G ad harmoic mea H satisfy the relatio G + 3 H = 8. Fid the two umbers. 7. If betwee ay two quatities there be iserted two arithmetic meas A, A ; two geometric meas G, G ; ad two harmoic meas H, H the prove that G G : H H = A + A : H + H. 8. If 9 AMs ad agai 9 HMs are iserted betwee ad 3 the prove that A + 6 H correspodig HM. = 5, A is ay AM ad H the 9. Usig the relatio A.M. ³ G.M. prove that (i) ta q + cot q ³ ; if 0 < q < p (ii) (iii) (x y + y z + z x) (xy + yz + zx ) ³ 9x y z. (x, y, z are positive real umber) (a + b). (b + c). (c + a) ³ abc ; if a, b, c are positive real umbers 30. If a, b, c are positive real umbers the prove that b c + c a + a b ³ abc (a + b + c). 3. Fid the sum of the terms of the series whose th term is (i) ( + ) (ii) 3 3. Fid the sum to -terms of the sequece. (i) to -terms. (ii) to terms. 33. Fid the sum i the th group of sequece, (i), (, 3); (, 5, 6, 7); (8, 9,..., 5);... (ii) (), (, 3, ), (5, 6, 7, 8, 9), Fid the sum to -terms of the sequece. (i) (ii) Sum the followig series to terms. (i) (ii) å r = r (r + ) (r + ) (r + 3) ( + )( + ) 36. Sum of the followig series (i) (ii) If a, b, c are positive real umbers ad sides of the triagle the prove that (a + b + c) 3 ³ 7 (a + b c) (c + a b) (b + c a) Arride learig Olie E-learig Academy A-79 Idra Vihar, Kota Rajastha 3005 Page No. # 3

15 PART-I IIT-JEE (PREVIOUS YEARS PROBLEMS). If a, a, a 3,..., a are positive real umbers whose product is a fixed umber c, the the miimum value of a + a + a a + a is [IIT - 00, 3] (A) (c) / (B) ( + ) c / (C) c / (D) ( + )(c) /. Suppose a, b, c are i A.P. ad a, b, c are i G.P. if a < b < c ad a + b + c = 3, the the value of a is [IIT - 00, 3] (A) (B) 3 (C) 3 (D) 3. Let a, b be positive real umbers. If a, A, A, b are i arithmetic progressio,a, G, G, b are i geometric progressio ad a, H, H, b are i harmoic progressio, show that G G H H = A + A H + H = ( a + b )( a + b ). [IIT 00, 5 ] 9ab. If a Î æ 0, p ö ç the ta a x + x + è ø x + x is always greater tha or equal to: [IIT- 003, 3] (A) ta a (B) (C) (D) sec a 5. If a, b & c are i arithmetic progressio ad a, b & c are i harmoic progressio, the prove that either a = b = c or a, b & - c are i geometric progressio. [IIT 003, ] 6. A ifiite G.P. has first term as x ad sum upto ifiity as 5. The the rage of values of x is: [IIT - 00, 3] (A) x 0 (B) x ³ 0 (C) 0 < x < 0 (D) 0 x 0 7. I the quadratic equatio ax + bx + c = 0, D = b ac ad a + b, a + b, a 3 + b 3, are i G.P. where a, b are the root of ax + bx + c = 0, the [IIT - 005] (A) D ¹ 0 (B) bd = 0 (C) cd = 0 (D) D = æ 3 ö æ 3 ö æ 3 ö 8. If a = ç ø + ç ø +...( ) è è ç ø ad b è = a, the fid the miimum atural umber 0 such that b > a " > 0 [IIT 006, 6] Arride learig Olie E-learig Academy A-79 Idra Vihar, Kota Rajastha 3005 Page No. #

16 Comprehesio [IIT - 007] Let V r deote the sum of the first r terms of a arithmetic progressio (A.P.) whose first term is r ad the commo differece is (r ). Let T r = V r + V r ad Q r = T r + T r for r =,, The sum V + V V is (A) ( + ) (3 + ) (B) ( + ) (3 + + ) (C) ( + ) (D) 3 ( 3 + 3) 0. T r is always (A) a odd umber (C) a prime umber (B) a eve umber (D) a composite umber. Which oe of the followig is a correct statemet? (A) Q, Q, Q 3,... are i A.P. with commo differece 5 (B) Q, Q, Q 3,... are i A.P. with commo differece 6 (C) Q, Q, Q 3,... are i A.P. with commo differece (D) Q = Q = Q 3 =... Comprehesio [IIT - 007] Let A, G, H deote the arithmetic, geometric ad harmoic meas, respectively, of two distict positive umbers. For ³, let A ad H have arithmetic, geometric ad harmoic meas as A, G, H respectively.. Which oe of the followig statemets is correct? (A) G > G > G 3 >... (B) G < G < G 3 <... (C) G = G = G 3 =... (D) G < G 3 < G 5 <... ad G > G > G 6 > Which oe of the followig statemets is correct? (A) A > A > A 3 >... (B) A < A < A 3 <... (C) A > A 3 > A 5 >... ad A < A < A 6 <... (D) A < A 3 < A 5 <... ad A > A > A 6 >.... Which oe of the followig statemets is correct? (A) H > H > H 3 >... (B) H < H < H 3 <... (C) H > H 3 > H 5 >... ad H < H < H 6 <... (D) H < H 3 < H 5 <... ad H > H > H 6 > Suppose four distict positive umbers a, a, a 3, a are i G.P. Let b = a, b = b + a, b 3 = b + a 3 ad b = b 3 + a. Statemet - : The umbers b, b, b 3, b are either i A.P. or i G.P. ad Statemet - : The umbers b, b, b 3, b are i H.P. Arride learig Olie E-learig Academy A-79 Idra Vihar, Kota Rajastha 3005 Page No. # 5

17 (A) STATEMENT- is True, STATEMENT- is True ; STATEMENT- is a correct explaatio for STATEMENT- (B) STATEMENT- is True, STATEMENT- is True ; STATEMENT- is NOT a correct explaatio for STATEMENT- (C) STATEMENT- is True, STATEMENT- is False (D) STATEMENT- is False, STATEMENT- is True [IIT-JEE 008, Paper-, (3, ), 8] 6. If the sum of first terms of a A.P. is c, the the sum of squares of these terms is : [IIT-JEE - 009, Paper-, (3, ), 80] (A) - ( ) c 6 (B) + ( ) c 3 (C) - ( ) c 3 (D) + ( ) c 6 6 p æ (m-) pö æ mpö 7.* For 0 < q <, the solutio(s) of å cosec ç q+ cosec q+ m = ç = is(are) : è ø è ø [IIT-JEE - 009, Paper-, (, ), 80] (A) p (B) 6 p (C) p 5p (D) 8. Let S k, k =,,..., 00, deote the sum of the ifiite geometric series whose first term is k k! ad the commo ratio is 00 k. The the value of 00 00! + å (k 3k+ ) Sk is. [IIT-JEE - 00, Paper-, (3, 0), 8] k= 9. Let a, a, a 3,..., a be real umbers satisfyig a = 5, 7 a > 0 ad a k = a k a k for k = 3,,...,. If a + a a = 90, the the value of a + a a is equal to. [IIT-JEE - 00, Paper-, (3, 0), 79] 0. Let a, a, a 3,..., a 00 be a arithmetic progressio with a = 3 ad S p = p å ai, p 00. i = For ay iteger with 0, let m = 5. If S S m does ot deped o, the a is. [IIT-JEE - 0, Paper-, (3, 0), 80]. The miimum value of the sum of real umbers a 5, a, 3a 3,, a 8 ad a 0 with a > 0 is. [IIT-JEE - 0, Paper-, (3, 0), 80]. Let a, a, a 3,... be i harmoic progressio with a = 5 ad a 0 = 5. The least positive iteger for which a < 0 is : (A) (B) 3 (C) (D) 5 [IIT-JEE 0, Paper-, (3, ), 66] Arride learig Olie E-learig Academy A-79 Idra Vihar, Kota Rajastha 3005 Page No. # 6

18 PART-II AIEEE (PREVIOUS YEARS PROBLEMS). The sum of the series = [AIEEE 00] () 300 () 5 (3) 5 () 0. If the sum of a ifiite GP is 0 ad sum of their square is 00 the commo ratio will be = [AIEEE 00] () / () / (3) 3/5 () 3. If the third term of a A.P. is 7 ad its 7th term is more tha three times of its 3rd term, the sum of its first 0 terms is : [AIEEE 00] () 8 () 7 (3) 70 () 090. If x, x, x 3 ad y,y, y 3 are both i GP with the same commo ratio, the the poits (x, y ), (x,y ) ad (x 3,y 3 ) : [AIEEE 003] () lie o a straight lie () lie o a elipse (3) lie o a circle () are vertices of a triagle 5. Let T r be the rth term of a AP whose first term is a ad commo differece is d. If for some positive itegers m &, m ¹, T m = ad T = m, the a d equals : [AIEEE 00] () 0 () (3) m () + m 6. If x = å =0 a, y = å =0 b, z = å =0 c where a,b,c are i AP ad a <, b <, c <, the x,y,z are i : () HP () Arithmetico Geometric Progressio (3) AP () GP [AIEEE 005] 7. If i a DABC, the altiudes from the vertices A, B, C o opposite sides are i H.P., the si A, si B, si C are i : [AIEEE 005] () G.P. () A.P. (3) Arithmetico-Geometirc progressio () H.P. 8. Let a, a, a 3,... be terms of a AP. If () 7 () 7 a + a a + a a a p q (3) p a6, p ¹ q, the equals : [AIEEE 006] a = q () 9. If a, a,..., a are i HP, the the expressio a a + a a a a is equal to : [AIEEE 006] () ( ) (a a ) () a a (3) ( ) a a () (a a ) 0. I a geometric progressio cosistig of positive terms, each term equals the sum of the ext two terms. The the commo ratio of this progressio equals : [AIEEE 007] () ( - 5) () 5 (3) 5 () ( 5 -). A perso is to cout 500 currecy otes. Let a deote the umber of otes he couts i the th miute. If a = a =...= a 0 = 50 ad a 0, a,...are i a AP with commo differece, the the time take by him to cout all otes is : [AIEEE 00] () 3 miutes () 5 miutes (3) 35 miutes () miutes. The sum of first 0 terms of the sequece 0.7, 0.77, 0.777,..., is : [JEE Mais 03] () 7 ( ) () ( ) 0 (3) ( ) () ( ) Arride learig Olie E-learig Academy A-79 Idra Vihar, Kota Rajastha 3005 Page No. # 7

19 NCERT BOARD QUESTIONS. Fid the umber of terms i the series The sum of three cosecutive terms of a icreasig A.P. is 5. If the product of the first ad third of these terms be 73, the fid third term 3. If we divide 0 ito four parts which are i A.P. such that product of the first ad the fourth is to the product of the secod ad third is the same as : 3, the fid the smallest part. I ay G.P. the first term is ad last term is 5 ad commo ratio is, the fid 5 th term from ed 5. Break the umbers 55 ito three parts so that the obtaied umbers form a G.P., the first term beig less tha the third oe by 0-6. A ball falls from a height of 00 mts. o a floor. If i each reboud it describes /5 height of the previous fallig height, the fid the total distace travelled by the ball before comig to rest 7. Fid the sum of 0 terms of the series I a potato race, 8 potatoes are placed 6 meters apart o a straight lie, the first beig 6 meters from the basket which is also placed i the same lie. A cotestat starts from the basket ad puts oe potato at a time ito the basket. Fid the total distace he must ru i order to fiish the race : 9. The sum of terms of two airthmetic series i the ratio of (7 + ) : ( + 7). Fid the ratio of their th term. 0. Sum of the series to terms ad to ifiity The first term of a A.P. is a, ad the sum of the first p terms is zero, show that the sum of its ext q terms is - a(p + q)q. p-. A ma saved Rs i 0 years. I each succeedig year after the first year he saved Rs 00 more tha what he saved i the previous year. How much did he save i the first year? 3. A ma accepts a positio with a iitial salary of Rs 500 per moth. It is uderstood that he will receive a automatic icrease of Rs 30 i the very ext moth ad each moth ad each moth thereafter. (a) Fid his salary for the teth moth (b) What is his total earigs durig the first year? Arride learig Olie E-learig Academy A-79 Idra Vihar, Kota Rajastha 3005 Page No. # 8

20 p æq ö. If the pth ad qth terms of a G.P. are q ad p respectively, show that its (p + q) th term is ç q èp ø 5. A carpeter was hired to build 9 widow frames. The first day he made five frames ad each day, thereafter he made two more frames tha he made the day before. How may days did it take him to fiish the job? p-q 6. We kow the sum of the iterior agles of a triagle is 80º. Show that the sums of the iterior agles of polygos with 3,, 5, 6,... sides form a arithmetic progressio. Fid the sum of the iterior agles for a sided polygo. 7. A side of a equilateral triagle is 0cm log. A secod equilateral triagle is iscribed i it by joiig the mid poits of the sides of the first triagle is iscribed i it by joiig the mid poits of the sides of the first triagle. The process is cotiued as show i the accompayig diagram. Fid the perimeter of the sixth iscribed equilateral triagle. 8. I a cricket touramet 6 school teams participated. A sum of Rs 8000 is to be awarded amog themselves as prize moey. If the last placed team is awarded Rs 75 i prize moey ad the award icreases by the same amout for successive fiishig places, how much amout will the first place team receive? 9. If a, a, a 3,... a are i A.P., where a i > 0 for all i, show that = a + a a + a a - + a a + a 3 0. Fid the sum of the series (3 3 3 ) + (5 3 3 ) + ( ) +... to (i) terms (ii) 0 terms. Fid the r th term of a A.P. sum of whose first terms is If A is the arithmetic mea ad G, G be two geometric meas betwee ay two umbers, the prove that A = G G G + G 3. If q, q, q 3..., q are i A.P., whose commo differece is d. show that sec q sec q + sec q sec q sec q secq = taq -taq sid. If the sum of p terms of a A.P. is q ad the sum of q terms is p. show that the sum of p + q terms is (p + q). Also, fid the sum of first p q terms (p > q). 5. If p th, q th, ad r th terms of a A.P. ad G.P. are both a, b ad c respectively, show that a b c. b c a. c a b = Arride learig Olie E-learig Academy A-79 Idra Vihar, Kota Rajastha 3005 Page No. # 9

21 EXERCISE # PART # I A-. (B) A-. (B) A-3. (D) A-. (C) A-5. (D) A-6. (B) A-7. (A) A-8. (B) A-9. (A) A-0. (C) A-*. (B, D) B-. (B) B-. (D) B-3. (C) B-. (C) B-5*. (A, C) B-6*. (B, C) C-. (C) C-*. (A, B) C-3. (C) C-. (C) C-5. (C) C-6. (A) C-7. (C) C-8. (B) C-9*. (A,B,C) C-0. (B) C-*. (A, B, C, D) C-. (C) PART # II. (A). (A) 3. (D). (B) 5. (C) 6. (C) 7. (B) 8. (A) 9. (A- p), (B - p), (C - q), (D - r) 0. (A- q), (B - s), (C - r), (D - p). (A). (A) 3. (C) EXERCISE # PART # I. (C). (C) 3. (C). (D) 5. (A) 6. (A) 7. (C) 8. (C) 9. (D) 0. (A). (C). (C) 3. (A). (A) 5. (C) 6*. (B, C) 7. (A, B, C, D) PART # II., 5, 8, , , 9,. 8., 6, 8. 3, 7, or, 7, 5. q-r p-q 6. 6, 3, 3/, pr ; R 9. ( ); (/3 + 5/3 + 65/6 +...) 0. p, p p, A = 3; B = (i) - (ii) 3 6. a =, b = Arride learig Olie E-learig Academy A-79 Idra Vihar, Kota Rajastha 3005 Page No. # 0

22 3. (i) ( + ) ( + 7) (ii) (3 + + ) (i) + 3 (ii) 7 ( ) 33. (i) - ( ) (ii) ( - ) (i) ( + )( + 3) (ii) ( + ) ( + ) ( + 3) ( + 3) (i) (/5) ( + ) ( + ) ( + 3) ( + ) (ii) ( + ) ( + ) 36. (i) 5 5 (ii) ( + ) ; s = ( + + ) EXERCISE # 3 PART # I. (A). (D). (A) 6. (C) 7. (C) 8. miimum atural umber 0 = 6 9. (B) 0. (D). (B). (C) 3. (A). (B) 5. (C) 6. (C) 7.* (C, D) , (D) PART # II. (3). (3) 3. (3). () 5. () 6. () 7. () 8. (3) 9. (3) 0. (). (3) EXERCISE # ,5, mts F I 89 + HG 0 K J ( 6) / (8 + 3) (3 + ) (3 + ),. Rs00 3. Rs 8080, Rs days 6. 30º 7. 5 cm 8 8. Rs (i) (ii) 960. T r = 6r Arride learig Olie E-learig Academy A-79 Idra Vihar, Kota Rajastha 3005 Page No. #