sum of terms problem - please help explain

kjallow wrote:Question: For every integer K from 1-10, inclusive, the Kth term of a certain sequence 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

I tried using the formula below but just don't get it
Sum = average * # of terms
where average = (1st term + last term)/2

official answer is d - between 1/4 and 1/2

Anurag@Gurome wrote:

kjallow wrote:Question: For every integer K from 1-10, inclusive, the Kth term of a certain sequence 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

I tried using the formula below but just don't get it
Sum = average * # of terms
where average = (1st term + last term)/2

official answer is d - between 1/4 and 1/2

We know that the sum of n terms of a geometric series is given by:
S(n) = a(1 - r�)(1 - r), where a is the first term, r is the common ratio of the geometric progression and n = number of terms.

Here, a = 1/2, r = -1/2, n = 10
T = 1/2[1 - (-1/2)^10]/[1 + 1/2]
= 1/2[1 - 1/1024]/[3/2]
= 1/2 * 1023/1024 * (2/3)
= (1023/1024) * (1/3)
Now (1023/1024) = 1 approx, so T = 1/3, which lies between 1/4 and 1/2.

The correct answer is D.