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
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
Give examples of population explosion. Ask students for examples.
General Term
an = ar(n-1)

4. _I005 Choose Well!!!! Problem Solving Tip # 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
a, b, c are in G.P.  b2 = ac

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

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

8. Illustration
_I007
Consider a rectangle. Add a rectangle of half the area to it. Repeat this process infinite times.We can see that eventually we will have a rectangle twice the size of the original rectangle. This is as predicted by the formula a/(1-r) 

9. _I008 Single Geometric Mean G is the G.M. of a and b  G2 = ab
a, G, b are in G.P. means G is G.M. of a and b. Explain that G.M. of two numbers CAN be positive or negative, but it is CUSTOMARY to take the positive value if the numbers are positive and the negative value if the numbers are negative.

10. Geometric Mean – a Definition
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
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,
Product of n G.M.s between two numbers is the nth power of a single G.M. between the numbers.

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

14. _I008 Illustrative Problem
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)

15. _I008 Illustrative Problem A. Given that Squaring both sides, we get,
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,
Dividing by b2 and putting =r, we get,

16. _I008 Illustrative Problem A.  Ans : (a)
Q. If the A.M. between a and b is twice as great as the G.M., a:b is equal to A.
Ask students to explain why we have two values and what they signify. Which one will we choose?
 Ans : (a)

17. Arithmetico-Geometric Progression
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 is the product of general term of an A.P. and general term of a G.P.
General Term
an = {a+(n-1)d}r(n-1)

18. _I010 Sum of n Terms of an A.G.P. Consider an A.G.P. with general
term {a+(n-1)d}r(n-1).
Let the sum of first n terms be Sn
The formula derived here is of little practical value. It is normal to sum an A.G.P. using the method used in the derivation.

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

20. Sum of Infinite Terms of an A.G.P.
Sum of n terms of an A.G.P.,
a+(a+d)r+(a+2d)r2……..

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

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

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

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

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

26. _I012 Harmonic Progression
A sequence is called a harmonic progression (H.P.) if the reciprocals of its terms form an A.P.
First term
General Term
It is usual to solve problems in H.P. by taking reciprocal of each term and then solving for the resulting A.P.can any term of A.P. be 0

27. 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 :

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

30. _I007 Class Exercise Q2. A. S for the given G.P.
Q. Find the value of p, if S for the G.P.
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. _I008 Class Exercise Q3. A. Let the two quantities be a and b.
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.
Common difference =
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. _I008 Class Exercise Q4. A. Common ratio Given that
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. Class Exercise Q5.
_I010
Q. Find the sum of the series :

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

37. _I010 Class Exercise Q5. Q. Find the sum of the series :
Please check the calculation carefully

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

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

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

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

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

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

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. _I012 Class Exercise Q9 A. (iv)-(v), (v)-(vi), (vi)-(iv) gives,
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. _I012 Class Exercise Q9 A. (vii)c, (viii)a and (ix)b gives, Adding,
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. _I012 Class Exercise Q9 A.  (q-r)bc+(r-p)ca+(p-q)ab = 0  Ans : (c)
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

51. _I012 Class Exercise Q10. A. ax = by = cz =  (say)
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
A. ax = by = cz =  (say)
x, y, z are in H.P.
 Ans : (c)