For every integer k from 1 to 10, inclusive, the kth term of a sequence is given by (-1)raised to k+1 (1/2 raised to k). If T is the sum of the first 10 terms of the sequence, then T is
A . greater than 2
B. between 1 and 2
C. between ½ and 1
D. between ¼ and ½
E. less than ¼
Pls explain the answer!!
You could learn a formula for such questions (this is an alternating geometric series), but you don't need to. It's almost always a good idea to write out at least the first few terms of a sequence- often you'll notice a pattern that will help to answer the question. If you do this, you'll see that the question is asking:
1/2 - 1/4 + 1/8 -1/16 + ... + 1/(2^9) - 1/2^(10) = ?
There are many ways to estimate the value of this sum. You can notice, for example, that 1/2 - 1/4 = 1/4; 1/8 - 1/16 = 1/16, and so on, so the sum is exactly equal to:
1/4 + 1/16 + 1/64 + 1/256 + 1/1028
and which is clearly only slightly greater than 1/4.
There's another way you could look at this, though it becomes much more clear when you can draw everything on a number line, which I can't do here. Label the points 0 and 1 on a number line. Suppose you're going to do the following:
-run halfway from 0 to 1, arriving at A: then A is 1/2
-run backwards half the distance between A and 0, arriving at B: then B is 1/2 - 1/4
-run forwards half the distance between B and A, arriving at C: then C is 1/2 - 1/4 + 1/8
-run backwards half the distance between C and B, arriving at D: then D is 1/2 - 1/4 + 1/8 - 1/16
you'll see that no matter how many terms you add of this sequence, the sum must be between 1/4 and 1/2.
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