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value of T

by nikhilsrl » Sun Mar 20, 2011 2:51 am
For every integer K from 1 to 10, kth term of a certain sequence can be written as [(-1)^(k+1)]*(1/2^k). If T is the sum of 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

OA is D

I want to know how to solve for this.
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by Anurag@Gurome » Sun Mar 20, 2011 3:35 am
nikhilsrl wrote:For every integer K from 1 to 10, kth term of a certain sequence can be written as [(-1)^(k+1)]*(1/2^k). If T is the sum of 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
1st term : [(-1)^(1 + 1)]*(1/2^1) = 1/2
2nd term : [(-1)^(2 + 1)]*(1/2^2) = -1/4
3rd term : [(-1)^(3 + 1)]*(1/2^3) = 1/8

Hence, the terms are 1/2, -1/4, 1/8, -1/16, ... etc

Now, we see that the first term is 1/2 and after that we alternately subtract and add half of the previous term. Hence there is no way the sum of the terms will ever be greater than 1/2. So first three options are discarded.

Now, note that sum of first two terms is 1/4 and after that we alternately add and subtract half of the previous term. Hence we make the sum greater than 1/4 by 1/8 and then decrease it by 1/16 and so on. Hence the sum is never going to be less than 1/4. So last option is also discarded.

The correct answer is D.
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