Series Problem

[JeetGulia]

For Every integer K from 1 to 10, Inclusive, the Kth Term is given by (-1)^k+1(1/2^k). If T is the sum of the first 10 terms in the sequence, then T is

A) greater than 2
B) Between 1 and 2
c) between 1/2 and 1
d) Between 1/4 and 1/2
e) less than 1/4

Ans D Corrected

[Rahul@gurome]

Kth term = (-1)^k+1 * (1/2^k)
1st term = (-1)^1+1 * (1/2^1) = 1/2
2nd term = (-1)^2+1 * (1/2^2) = -1/2^2
So, T = 1/2 - 1/2^2 + 1/2^3 - 1/2^4 +... up to 10 terms
T = (1/2 + 1/2^3 + 1/2^5 + ....) - (1/2^2 + 1/2^4 +....) = (1/2)[1 + 1/2^2 + 1/2^4 + ....] - (1/2^2)[1 + 1/2^2 + 1/2^4 +....]
= (1/2 - 1/4)[1 + 1/2^2 + 1/2^4 + ....] = [1/4][1 + 1/2^2 + 1/2^4 + ....], which implies T > 1/4 and 1 + 1/2^2 + 1/2^4 + .... will always be less than 2.
So, 1/4 < T < 1/2

Please check options (C) and (D), both of them are the same.

[klmehta03]

I agree the ans is D

[klmehta03]

I agree the ans is D