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

relaxin99 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

= 1/2 -1/2^2 +1/2^3-1/2^4.... -1/2^10
= [1/2+1/2^3+1/2^5+1/2^7+1/2^9]-1/2[1/2+1/2^3+1/2^5+1/2^7+1/2^9]
=1/2*[1/2+1/2^3+1/2^5+1/2^7+1/2^9]
= 1/2* [Value between 1/2 and 1]
= Value between 1/4 and 1/2

IMO - D

Think of it in two parts.

The first part (-1)^(k+1) is going to give you a positive 1 for all odd K terms, and a negative K for all even K terms, ie k=2 = -1^(2+1) = -1, while k=1 = -1^(1+1) = 1

The second part creates fractions based on 2. k=1 = 2^(-1*1) = 2^(-1) or 1/(2^1) = .5, k=2 = 2^(-1*2) = 2^(-2) or 1/(2^2) = .25

(1*.5)+(-1*.25)..... = approx 1/3 or D