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Sequences and Series

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  1. Session MPTCP05 Sequences and Series

  2. Session Objectives • Revisit G.P. and sum of n terms of a G.P. • Sum of infinite terms of a G.P. • Geometric Mean (G.M.) and insertion of n G.M.s between two given numbers • Arithmetico-Geometric Progression (A.G.P.) - definition, nth term • Sum of n terms of an A.G.P. • Sum of infinite terms of an A.G.P. • Harmonic Progression (H.P.) - definition, nth term

  3. Geometric Progression _I005 A sequence is called a geometric progression (G.P.) if the ratio between any term and the previous term is constant. The constant ratio, generally denoted by r is called the common ratio. a1 = a a2 = ar a3 = ar2 a4 = ar3 First term General Term an = ar(n-1)

  4. Problem Solving Tip _I005 Choose Well!!!! # Terms Common ratio 3 a/r, a, ar r 4 a/r3, a/r, ar, ar3 r2 5 a/r2, a/r, a, ar, ar2 r 6 a/r5, a/r3, a/r, ar, ar3, ar5 r2

  5. Important Properties of G.P.s _I005 a, b, c are in G.P.  b2 = ac

  6. Sum of n Terms of a G.P. _I006 Sn = a+ar+ar2+ar3+ . . .+ar(n-1) ………(i) Multiplying by r, we get, rSn = ar+ar2+ar3+ . . .+ar(n-1)+arn ……...(ii) Subtracting (i) from (ii), (r-1)Sn = a(rn-1)

  7. Sum of Infinite Terms of a G.P. _I007 Sum of n terms of a G.P.,

  8. Illustration _I007 

  9. Single Geometric Mean _I008 G is the G.M. of a and b  G2 = ab

  10. Geometric Mean – a Definition _I008 If n terms G1, G2, G3, . . . Gn are inserted between two numbers a and b such that a, G1, G2, G3, . . . , Gn, b form a G.P., then G1, G2, G3, . . . , Gn are called geometric means (G.M.s) of a and b.

  11. Geometric Mean – Common Ratio _I008 Let n G.M.s be inserted between two numbers a and b The G.P. thus formed will have (n+2) terms. Let the common ratio be r Now b = ar(n+2-1) = ar(n+1)

  12. Property of G.M.s _I008 Let n G.M.s G1, G2, G3, . . ., Gn be inserted between a and b. Then,

  13. A. Let the required G.M.s be G1, G2 and G3. Common ratio r = Illustrative Problem _I008 Q. Insert 3 G.M.s between 4 and 9

  14. Q. If the A.M. between a and b is twice as great as the G.M., a:b is equal to (a) (b) (c) (d) Illustrative Problem _I008

  15. Dividing by b2 and putting =r, we get, Illustrative Problem _I008 Q. If the A.M. between a and b is twice as great as the G.M., a:b is equal to A. Given that Squaring both sides, we get,

  16. A. Illustrative Problem _I008 Q. If the A.M. between a and b is twice as great as the G.M., a:b is equal to  Ans : (a)

  17. Arithmetico-Geometric Progression _I009 A sequence is called an arithmetico-geometric progression (A.G.P.) if the nth term is a product of the nth term of an A.P. and the nth term of a G.P. a1 = a a2 = (a+d)r a3 = (a+2d)r2 a4 = (a+3d)ar3 First term General Term an = {a+(n-1)d}r(n-1)

  18. Sum of n Terms of an A.G.P. _I010 Consider an A.G.P. with general term {a+(n-1)d}r(n-1). Let the sum of first n terms be Sn

  19. Illustrative Problem _I010 Q. Find the sum of the first 10 terms of the given sequence : 1, 3x, 5x2, 7x3, . . . A. Let S = 1+3x+5x2+7x3+ . . . +{1+(10-1)2}x(10-1) S = 1+3x+5x2+7x3+ . . . +19x9 xS = x+3x2+5x3+ . . . +17x9 +19x10 S-xS = 1+(2x+2x2+2x3+ . . . 2x9)-19x10

  20. Sum of Infinite Terms of an A.G.P. _I011 Sum of n terms of an A.G.P.,

  21. Sum of Infinite Terms of an A.G.P. _I011 Sum of n terms of an A.G.P.,

  22. Q. The sum to infinity of the series is (a) 16/35 (b) 11/8 (c) 35/16 (d) 8/6 Illustrative Problem _I011

  23. Q. The sum to infinity of the series is Illustrative Problem _I011 A. Let the required sum be S  Ans : (c)

  24. Q. The sum of the infinite series 1 + (1+b)r + (1+b+b2)r2 + (1+b+b2+b3)r3 . . ., r and b being proper fractions is : Illustrative Problem _I011

  25. Illustrative Problem _I011 Q. The sum of the infinite series 1 + (1+b)r + (1+b+b2)r2 + (1+b+b2+b3)r3 . . . (r and b being proper fractions ) is : A. Let the required sum be S Subtracting, we have,  Ans : (a)

  26. Harmonic Progression _I012 A sequence is called a harmonic progression (H.P.) if the reciprocals of its terms form an A.P. First term General Term

  27. Q. The first two terms of an infinite G.P. are together equal to 5, and every term is 3 times the sum of all the terms that follow it, the series is : Class Exercise Q1. _I007

  28. Class Exercise Q1. _I007 Q. The first two terms of an infinite G.P. are together equal to 5, and every term is 3 times the sum of all the terms that follow it, the series is : A. Let the first term of the G.P. be a and the common ratio be r. Given that a+ar = 5 and Now,  Ans : (a)

  29. Q. Find the value of p, if S for the G.P. Class Exercise Q2. _I007

  30. Q. Find the value of p, if S for the G.P. Class Exercise Q2. _I007 A. S for the given G.P.

  31. Class Exercise Q3. _I008 Q. If one G.M. G and two A.M.s p and q are inserted between two quantities, show that G2 = (2p-q)(2q-p).

  32. Common difference = Class Exercise Q3. _I008 Q. If one G.M. G and two A.M.s p and q are inserted between two quantities, show that G2 = (2p-q)(2q-p). A. Let the two quantities be a and b. a, p, q, b are in A.P. Q.E.D.

  33. Class Exercise Q4. _I008 Q. n G.M.s are inserted between 16/27 and 243/16. If the ratio of the (n-1)th G.M. to the 4th G.M. is 9 : 4, find n.

  34. Class Exercise Q4. _I008 Q. n G.M.s are inserted between 16/27 and 243/16. If the ratio of the (n-1)th G.M. to the 4th G.M. is 9 : 4, find n. A. Common ratio Given that

  35. Q. Find the sum of the series : Class Exercise Q5. _I010

  36. Q. Find the sum of the series : Class Exercise Q5. _I010 A. We see that

  37. Q. Find the sum of the series : Class Exercise Q5. _I010

  38. Class Exercise Q6. _I010 Q. Find sum to n terms of the series : 1+2x+3x2+4x3+ . . . (x  1)

  39. Class Exercise Q6. _I010 Q. Find sum to n terms of the series : 1+2x+3x2+4x3+ . . . (x  1) A. We see that an = nxn-1  Sn = 1+2x+3x2+4x3+ . . . +nxn-1  xSn = x+2x2+3x3+. . . +(n-1)xn-1+nxn  (1-x)Sn = 1+(x+x2+x3+ . . . xn-1)-nxn

  40. Q. Find the sum of infinite terms of the series : Class Exercise Q7. _I011

  41. Q. Find the sum of infinite terms of the series : A. Class Exercise Q7. _I011

  42. Q. Find the sum of the series : Class Exercise Q8. _I011

  43. Q. Find the sum of the series : A. Class Exercise Q8. _I011

  44. Class Exercise Q9. _I012 Q. If the pth, qth and rth terms of an H.P. be a, b, c respectively, then (q-r)bc+(r-p)ca+(p-q)ab is equal to (a) 1 (b) -1 (c) 0 (d) None of these

  45. Class Exercise Q9. _I012 Q. If the pth, qth and rth terms of an H.P. be a, b, c respectively, then (q-r)bc+(r-p)ca+(p-q)ab is equal to (a) 1 (b) -1 (c) 0 (d) None of these A. The reciprocals the terms of the H.P. will be in A.P. Let this A.P. have first term  and common difference . Given that

  46. Class Exercise Q9. _I012 Q. If the pth, qth and rth terms of an H.P. be a, b, c respectively, then (q-r)bc+(r-p)ca+(p-q)ab is equal to (a) 1 (b) -1 (c) 0 (d) None of these A. Taking reciprocal of (i), (ii) and (iii), we have

  47. Class Exercise Q9 _I012 Q. If the pth, qth and rth terms of an H.P. be a, b, c respectively, then (q-r)bc+(r-p)ca+(p-q)ab is equal to (a) 1 (b) -1 (c) 0 (d) None of these A. (iv)-(v), (v)-(vi), (vi)-(iv) gives,

  48. Class Exercise Q9 _I012 Q. If the pth, qth and rth terms of an H.P. be a, b, c respectively, then (q-r)bc+(r-p)ca+(p-q)ab is equal to (a) 1 (b) -1 (c) 0 (d) None of these A. (vii)c, (viii)a and (ix)b gives, Adding,

  49. Class Exercise Q9 _I012 Q. If the pth, qth and rth terms of an H.P. be a, b, c respectively, then (q-r)bc+(r-p)ca+(p-q)ab is equal to (a) 1 (b) -1 (c) 0 (d) None of these A.  (q-r)bc+(r-p)ca+(p-q)ab = 0  Ans : (c)

  50. Class Exercise Q10. _I012 Q. If ax = by = cz and x, y, z are in H.P. then a, b, c are in (a) A.P. (b) H.P. (c) G.P. (d) None of these