Problem 7 - Entrance Test

The sum of an infinite series 1 + (1/2) + (1/4) + (1/8) + ... is S. A new series is formed by taking every second term of this series (i.e., 1 + 1/4 + 1/16 + ...). The ratio of the sum of the new series to S is:

A. 1/3B. 2/3C. 1/2D. 3/4

Correct: B

The original series is a geometric series with first term a = 1 and common ratio r = 1/2. Sum S = a/(1-r) = 1/(1-1/2) = 2. The new series takes every second term: terms are 1, 1/4, 1/16, 1/64, ... This is also a geometric series with first term a' = 1 and common ratio r' = (1/4)/(1) = 1/4. (Each term is obtained by multiplying the previous term by 1/4.) Sum S' = a'/(1-r') = 1/(1-1/4) = 1/(3/4) = 4/3. Ratio S'/S = (4/3)/2 = (4/3)*(1/2) = 2/3. Therefore, the ratio is 2/3.