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 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
Can you guys show me how to solve this? thanks
I like this question. It takes something that's potentially hard and lets us draw a nice diagram that gets to the answer easily.
First, plot out how the sequence works. Very quickly, we will see that the sequence is made up of growing powers of 1/2, alternating positive and negative:
k = 1: 1/2
k = 2: -1/4
k = 3: 1/8
k = 4: -1/16
k = 5: 1/32
now, to add these quickly, draw a number line, with 0 in the middle, and +1 and -1 on the extremes.
The first term is 1/2, so start there. Then we subtract 1/4, which drops us down to 1/4. Then we add 1/8, so we go back up, but not quite to 1/2. Then subtract 1/16, we go down, but not exactly back to 1/4, then add 1/32, and on and on. We are essentially bouncing around a point that is in between 1/4 and 1/2.
And, lo and behold, that's it. The answer is D.