yvichman wrote:For every integer K from 1 to 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 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
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Dana is right in that you can use a formula to find the sum of the terms of a geometric sequence. However, we can also tackle it a different way.
First, get an idea of this sum. Find the first few terms to get:
T = 1/2 - 1/4 + 1/8 - 1/16 + . . .
We can rewrite this as T = (1/2 - 1/4) + (1/8 - 1/16) + . . .
Or T = 1/4 + 1/16 + 1/64 + 1/256 + 1/1024
Now examine the last 4 terms ( 1/16 + 1/64 + 1/256 + 1/1024)
Notice that 1/64, 1/256, and 1/1024 are each less than 1/16
So, (1/16 + 1/64 + 1/256 + 1/1024) < (1/16 + 1/16 + 1/16 + 1/16) = 1/4
Now start from the beginning: T = 1/4 + (1/16 + 1/64 + 1/256 + 1/1024), but since 1/16 + 1/64 + 1/256 + 1/1024 is less than 1/4 we know that T is less than 1/4 + 1/4 (1/2)
So, 1/4 < T < 1/2 (D)