Problem Solving

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

greater than 2
between 1 and 2
between 1/2 and 1
between 1/4 and 1/2
less than 1/4

It is not clear whether you mean (-1)^k + 1/2^k or (-1)^(k+1) * (1/2^k)?

The solution below assumes the former.
The sign changing terms (-1)^k are going to cancel out because you will have 5 that are +1 and 5 that are -1.

You are left with T = 1/2 + 1/2^2 + 1/2^3 + 1/2^4 + ... + 1/2^10

1/2 < T because the first term is 1/2 and you are adding positive numbers on top of it.

Draw the number axis and add graphically length segments corresponding to the different fractions in T = 1/2 + 1/4 + 1/8 + 1/16 + .... , the 1/2 segment starting from 0, each subsequent segment starting where the previous ends - the full length covered is a graphical visualization of the running total. Each subsequent fraction adds haft of the remaining segment to 1. You will see visually that the sum is approaching 1 from below when you increase the number of terms.

so 1/2 < T < 1

alex.gellatly 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 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

Notice that the answer choices are RANGES.
We are not expected to calculate the exact sum.
Use a NUMBER LINE to determine the correct range.



Follow the arrows.
The first term is 1/2.
When we add in -1/4 -- the second term -- the sum decreases to 1/4.
When we add in +1/8 -- the third term -- the sum increases to 3/8.
By now, we can already see that the sum will converge to a value somewhere between 1/4 and 3/8.

The correct answer is D.