Problem:
In a geometric sequence of real numbers, the sum of the first two terms is 7 and the sum of the first six terms is 91. The sum of the first four terms is
Answer Choices:
A. 28
B. 32
C. 35
D. 49
E. 84
Solution:
Let a and r be the first term and the common ratio of successive terms in the geometric sequence, respectively. Then
a+ar=7r−1a(r6−1)=91
Dividing the first equation into the second yields
(r2−1)(r6−1)r4+r2−12(r2+4)(r2−3)=13=0=0.
Thus r2=3 and
a+ar+ar2+ar3=(a+ar)(1+r2)=7(4)=28