Problem Solving

For every integer k from 1 to 10, inclusive , the kth term of certain sequence is given by [ (-1) ^ (k+1) ] [1/2^k]
If T is the sum of first 10 terms in the sequence , then T is

- greater than 2
- Between 1 and 2
- Between 1/2 and 1
- Between 1/4 and 1/2
- Less than 1/4

OA is [spoiler]Between 1/4 and 1/2[/spoiler]

When k = 1, f(k) = (-1)^(1+1) * (1/2)*1 = 1/2
When k = 2, f(k) = (-1)^(2+1) * (1/2)*2 = -1/4
When k = 3, f(k) = (-1)^(3+1) * (1/2)*3 = 1/8
and so on
The sequence is :
1/2 -1/4 +1/8 -1/16 + .....+ (1/2)^9 - (1/2)^10
or 1/4 + 1/16 + ... + 1/2^8 + 1/2^10 ( the multiplier is 1/4)
Now you see that the sequence is obviously larger than 1/4.
To prove the sequence is less than 1/2 is a little tougher.
Let S be the value of the sequence
S = 1/4 + 1/16 + ... + 1/2^8 + 1/2^10
S*4 = 1+ 1/4 + 1/16 + ...+1/2^8 (Note that the bold phrase is a part of the sequence)
S*4 = 1+ (1/4 + 1/16 + ...+1/2^8+1/2^10) - 1/2^10 ( add & subtract 1/2^10 at the same time to keep the sum unchanged)
S*4 = 1 + S - 1/2^10
3*S = 1 - 1/2^10
As 1/2^10 is very small, S*3 is near 1.
Thus S is something approximately to 1/3