Q.11 If S be the sum, P the product & R the sum of the reciprocals of a GP, find the value of

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1 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 ( up to terms) = l + l 3 Q3 There are AM s betwee & 3 such that 7th mea : ( ) th mea = 5 : 9, the fid the value of Q4 Fid the sum of the series, to terms Q5 Express the recurrig decimal 0576 as a ratioal umber usig cocept of ifiite geometric series Q6 Fid the sum of the terms of the sequece Q7 The first term of a arithmetic progressio is ad the sum of the first ie terms equal to 369 The first ad the ith term of a geometric progressio coicide with the first ad the ith term of the arithmetic progressio Fid the seveth term of the geometric progressio Q8 If the p th, q th & r th terms of a AP are i GP Show that the commo ratio of the GP is q r p q Q9 If oe AM a & two GM s p & q be iserted betwee ay two give umbers the show that p 3 + q 3 = apq Q0 The sum of terms of two arithmetic series are i the ratio of (7 + ) : (4 + 7) Fid the ratio of their th term Q If S be the sum, P the product & R the sum of the reciprocals of a GP, fid the value of R P S Q The first ad last terms of a AP are a ad b There are altogether ( + ) terms A ew series is formed by multiplyig each of the first terms by the ext term Show that the sum of the ew series is (4 )(a b ) (4 )ab 6 Q3 I a AP of which a is the Ist term, if the sum of the Ist p terms is equal to zero, show that the sum of the ext q terms is a (p + q) q/(p ) Q4(a) The iterior agles of a polygo are i AP The smallest agle is 0 & the commo differece is 5 Fid the umber of sides of the polygo The iterior agles of a covex polygo form a arithmetic progressio with a commo differece of 4 Determie the umber of sides of the polygo if its largest iterior agle is 7 Q5 A AP & a HP have the same first term, the same last term & the same umber of terms ; prove that the product of the r th term from the begiig i oe series & the r th term from the ed i the other is idepedet of r Q6 Fid three umbers a, b, c betwee & 8 such that ; (i) their sum is 5 (ii) the umbers, a, b are cosecutive terms of a AP & (iii) the umbers b, c, 8 are cosecutive terms of a GP Q7 Give that a x = b y = c z = d u & a, b, c, d are i GP, show that x, y, z, u are i HP Q8 I a set of four umbers, the first three are i GP & the last three are i AP, with commo differece 6 If the first umber is the same as the fourth, fid the four umbers

2 Brai Teasures Progressio ad Series By Abhijit kumar Jha 3 Q9 Fid the sum of the first terms of the sequece : 3 4 Q0 Fid the th term ad the sum to terms of the sequece : (i) (ii) Q The AM of two umbers exceeds their GM by 5 & HM by 7 Fid the umbers Q The harmoic mea of two umbers is 4 The airthmetic mea A & the geometric mea G satisfy the relatio A + G² = 7 Fid the two umbers Q3 Sum the followig series to terms ad to ifiity : (i) (iii) (ii) r 4 r Q4 Fid the value of the sum (a) r s i i j k r r (r + ) (r + ) (r + 3) 3 35 (iv) rs r 3 s where rs is zero if r s & rs is oe if r = s j Q5 For or 0 < < /, if : x = 0 cos, y = (i) xyz = xy + z 0 si, z = 0 (ii) xyz = x + y + z cos si the : Prove that EXERCISE II Q The series of atural umbers is divided ito groups (), (, 3, 4), (5, 6, 7, 8, 9), & so o Show that the sum of the umbers i the th group is ( ) Q The sum of the squares of three distict real umbers, which are i GP is S² If their sum is S, show that ² (/3, ) (, 3) Q3 If there be m AP s begiig with uity whose commo differece is,, 3 m Show that the sum of their th terms is (m/) (m m + + ) Q4 If S represets the sum to terms of a GP whose first term & commo ratio are a & r respectively, the prove that S + S 3 + S S - = a r a r r ( ) ( r) ( r) Q5 A geometrical & harmoic progressio have the same p th, q th & r th terms a, b, c respectively Show that a (b c) log a + b (c a) log b + c (a b) log c = 0 Q6 A computer solved several problems i successio The time it took the computer to solve each successive problem was the same umber of times smaller tha the time it took to solve the precedig problem How may problems were suggested to the computer if it spet 635 mi to solve all the problems except for the first, 7 mi to solve all the problems except for the last oe, ad 35 mi to solve all the problems except for the first two? Q7 If the sum of m terms of a AP is equal to the sum of either the ext terms or the ext p terms of the same AP prove that (m + ) [(/m) (/p)] = (m + p) [(/m) (/)] ( p)

3 Brai Teasures Progressio ad Series By Abhijit kumar Jha Q8 If the roots of 0x 3 cx 54x 7 = 0 are i harmoic progressio, the fid c & all the roots Q9(a) Let a, a a be a AP Prove that : = a a a a a a a a 3 a a a a a 3 a Show that i ay arithmetic progressio a, a a ² a ² + a 3 ² a 4 ² + + a² K a² K = [K/( K )] (a ² a² K ) Q0 Let a, a,, a, a +, be a AP Let S = a + a + a a S = a + + a a S 3 = a + + a a 3 Prove that the sequece S, S, S 3, is a arithmetic progressio whose commo differece is times the commo differece of the give progressio Q If a, b, c are i HP, b, c, d are i GP & c, d, e are i AP, Show that e = ab²/(a b)² Q If a, b, c, d, e be 5 umbers such that a, b, c are i AP ; b, c, d are i GP & c, d, e are i HP the: (i) Prove that a, c, e are i GP (ii) Prove that e = ( b a)²/a (iii) If a = & e = 8, fid all possible values of b, c, d Q3 The sequece a, a, a 98 satisfies the relatio a + = a + for =,, 3, 97 ad has the sum equal to 4949 Evaluate a k 49 k Q4 If is a root of the equatio x² ( ac) x (a² + c²) ( + ac) = 0 & if HM s are iserted betwee a & c, show that the differece betwee the first & the last mea is equal to ac(a c) Q5 (a) The value of x + y + z is 5 if a, x, y, z, b are i AP while the value of ; (/x)+(/y)+(/z) is 5/3 if a, x, y, z, b are i HP Fid a & b The values of xyz is 5/ or 8/5 accordig as the series a, x, y, z, b is a AP or HP Fid the values of a & b assumig them to be positive iteger Q6 A AP, a GP & a HP have a & b for their first two terms Show that their ( + ) th terms will be i GP if b ba a b a Q7 Prove that the sum of the ifiite series Q8 If there are quatities i GP with commo ratio r & S m deotes the sum of the first m terms, show that the sum of the products of these m terms take two & two together is [r/(r + )] [S m ] [S m ]

4 Brai Teasures Progressio ad Series By Abhijit kumar Jha Q9 Fid the coditio that the roots of the equatio x 3 px + qx r = 0 may be i AP ad hece solve the equatio x 3 x + 39x 8 = 0 Q0 If ax + bx + c = 0 & a x + b x + c = 0 have a commo root & a/a, b/b, c/c are i AP, show that a, b & c are i GP Q If a, b, c be i GP & log c a, log b c, log a b be i AP, the show that the commo differece of the AP must be 3/ Q If a = & for >, a = a - + Q3 Sum to terms : (i) (ii) a, the show that < a 75 < 5 x 3x x ( x ) ( x ) ( x ) ( x ) ( x 3) a a a 3 a a a a a a 3 Q4 I a GP the ratio of the sum of the first eleve terms to the sum of the last eleve terms is /8 ad the ratio of the sum of all the terms without the first ie to the sum of all the terms without the last ie is Fid the umber of terms i the GP Q5 Give a three digit umber whose digits are three successive terms of a GP If we subtract 79 from it, we get a umber writte by the same digits i the reverse order Now if we subtract four from the hudred's digit of the iitial umber ad leave the other digits uchaged, we get a umber whose digits are successive terms of a AP Fid the umber EXERCISE III Q For ay odd iteger, 3 ( ) ( ) l 3 = [ JEE 96, ] Q x = + 3a + 6a² + 0a 3 + a < y = + 4b + 0b² + 0b 3 + b <, fid S = + 3ab + 5(ab)² + i terms of x & y [ REE 96, 6 ] Q3 The real umbers x, x, x 3 satisfyig the equatio x 3 x² + x + = 0 are i AP Fid the itervals i which ad lie [JEE 96, 3] Q4 Let p & q be roots of the equatio x x + A = 0, ad let r & s be the roots of the equatio x 8x + B = 0 If p < q < r < s are i arithmatic progressio, the A =, ad B = [ JEE 97, ] Q5 a, b, c are the first three terms of a geometric series If the harmoic mea of a & b is ad that of b & c is 36, fid the first five terms of the series [ REE '98, 6 ] Q6 Select the correct alterative(s) [ JEE '98, ] (a) Let T r be the r th term of a AP, for r =,, 3, If for some positive itegers m, we have T m = & T = m, the T m equals : (A) m (B) (C) (D) 0 m If x =, y >, z > are i GP, the x, y, are i : z (A) AP (B) HP (C) GP (D) oe of the above (c) Prove that a triagle ABC is equilateral if & oly if ta A + ta B + ta C = 3 3

5 Brai Teasures Progressio ad Series By Abhijit kumar Jha Q7(a) The harmoic mea of the roots of the equatio 5 x 4 5 x = 0 is (A) (B) 4 (C) 6 (D) 8 Let a, a,, a 0, be i AP & h, h,, h 0 be i HP If a = h = & a 0 = h 0 = 3 the a 4 h 7 is: (A) (B) 3 (C) 5 (D) 6 [ JEE '99, + out of 00 ] Q8 The sum of a ifiite geometric series is 6 ad the sum of its first terms is 60 If the iverse of its commo ratio is a iteger, fid all possible values of the commo ratio, ad the first terms of the series [ REE '99, 6 ] Q9(a) Cosider a ifiite geometric series with first term 'a' ad commo ratio r If the sum is 4 ad the secod term is 3/4, the : (A) a = 7 4, r = 3 7 (B) a =, r = 3 8 (C) a = 3, r = (D) a = 3, r = 4 If a, b, c, d are positive real umbers such that a + b + c + d =, the M = (a + b) (c + d) satisfies the relatio : (A) 0 M (B) M (C) M 3 (D) 3 M 4 [ JEE 000, Screeig, + out of 35 ] (c) The fourth power of the commo differece of a arithmetic progressio with iteger etries added to the product of ay four cosecutive terms of it Prove that the resultig sum is the square of a iteger [ JEE 000, Mais, 4 out of 00 ] Q0 Give that, are roots of the equatio, A x 4 x + = 0 ad, the roots of the equatio, B x 6 x + = 0, fid values of A ad B, such that,, & are i HP [ REE 000, 5 out of 00 ] Q The sum of roots of the equatio ax + bx + c = 0 is equal to the sum of squares of their reciprocals Fid whether bc, ca ad ab i AP, GP or HP? [ REE 00, 3 out of 00 ] Q Solve the followig equatios for x ad y log x + log 4 x + log 6 x + = y ( 4y ) 3 5 ( y ) = 4log 4 x [ REE 00, 5 out of 00 ] Q3(a) Let be the roots of x x + p = 0 ad be the roots of x 4x + q = 0 If are i GP, the the itegral values of p ad q respectively, are (A), 3 (B), 3 (C) 6, 3 (D) 6, 3 (c) If the sum of the first terms of the AP, 5, 8, is equal to the sum of the first terms of the AP 57, 59, 6,, the equals (A) 0 (B) (C) (D) 3 Let the positive umbers a, b, c, d be i AP The abc, abd, acd, bcd are (A) NOT i AP/GP/HP (C) i GP (B) i AP (D) HP [ JEE 00, Screeig, + + out of 35 ]

6 Brai Teasures Progressio ad Series By Abhijit kumar Jha (d) Let a, a be positive real umbers i GP For each, let A, G, H, be respectively, the arithmetic mea, geometric mea ad harmoic mea of a, a, a Fid a expressio for the GM of G, G, G i terms of A, A A, H, H, H [ JEE 00 (Mais); 5] Q4(a) Suppose a, b, c are i AP ad a, b, c are i GP If a < b < c ad a + b + c = 3, the the value of a is (A) (B) 3 (C) (D) 3 [JEE 00 (Screeig), 3] Let a, b be positive real umbers If a, A, A, b are i AP ; a, a, a, b are i GP ad a, H, H, b are i HP, show that G G H H A H A H ( a b) ( a b) 9ab [ JEE 00, Mais, 5 out of 60 ] c Q5 If a, b, c are i AP, a, b, c are i HP, the prove that either a = b = c or a, b, form a GP [JEE-03, Mais-4 out of 60] Q6 The first term of a ifiite geometric progressio is x ad its sum is 5 The (A) 0 x 0 (B) 0 < x < 0 (C) 0 < x < 0 (D) x > 0 [JEE 004 (Screeig)] Q7 If a, b, c are positive real umbers, the prove that [( + a) ( + b) ( + c)] 7 > 7 7 a 4 b 4 c 4 [JEE 004, 4 out of 60] Q8(a) I the quadratic equatio ax + bx + c = 0, if = b 4ac ad +, +, are i GP where, are the roots of ax + bx + c = 0, the (A) 0 (B) b = 0 (C) c = 0 (D) = 0 [JEE 005 (Screeig)] If total umber of rus scored i matches is ( 4 + ) where >, ad the rus scored i the k th match are give by k + k, where k Fid [JEE 005 (Mais), ] Q9 If A ad B = A, the fid the miimum atural umber 0 such that B > A > 0 [JEE 006, 6]

7 Brai Teasures Progressio ad Series By Abhijit kumar Jha ANSWER KEY EXERCISE I Q Q 3 µ = 4 Q 4 S = (7/8){ } Q 5 35/ Q 6 ( + )/ (² + + ) Q 7 7 Q 0 (4 6)/(8 + 3) Q Q 4 (a) 9 ; Q 6 a = 5, b = 8, c = Q 8 (8, 4,, 8) Q 9 ² Q 0 (i) + 3 ; (ii) ² ; (/6) ( + ) ( + 3) + Q 0, 30 Q 6, 3 Q 3 (i) s = (/4) [/{6(3 + ) (3 + 4) }] ; s = /4 (ii) (/5) ( + ) ( + ) ( + 3) ( + 4) 35( )( ) (iii) /( + ) (iv) S = ; S 46()( ) = Q 4 (a) (6/5) (6 ) [ ( + ) ( + )]/6 EXERCISE II Q 6 8 problems, 75 miutes Q8 C = 9 ; (3, 3/, 3/5) Q (iii) b = 4, c = 6, d = 9 OR b =, c = 6, d = 8 Q3 499 Q 5 (a) a =, b = 9 OR b =, a = 9 ; a = ; b = 3 or vice versa Q9 p 3 9pq + 7r = 0; roots are, 4, 7 Q 3 (a) x ( x ) ( x ) ( x ) Q 4 = 38 Q 5 93 Q 4 ( ) ( + )² Q S = ab ( ab) ( a) ( a ) ( a ) EXERCISE III Where a = x /3 & b = y /4 Q3 (/3) ; (/7) Q 4 3, 77 Q 5 8, 4, 7, 6, 648 Q 6 (a) C B Q 7 (a) B D Q 8 r = ± /9 ; = ; a = 44/80 OR r = ± /3 ; = 4 ; a = 08 OR r = /8 ; = ; a = 60 Q 9 (a) D A Q 0 A = 3 ; B = 8 Q AP Q x = ad y = 3 Q 3 (a) A, C, (c) D, (d) A, A, A H, H, H Q4 (a) D Q6 B Q8 (a) C, = 7 Q9 0 = 5

EXERCISE - 01 CHECK YOUR GRASP

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