Problem: In a geometric series of positive terms the difference between the fifth and fourth terms is 576, and the difference between the second and first terms is 9. What is the sum of the first five terms of this series?
Answer Choices:
A. 1061
B. 1023
C. 1024
D. 768
E. none of these
Solution:
The sum of the first five terms of the geometric series with initial term a and common ratio r is
S5​=a+ar+ar2+ar3+ar4=1−ra(1−r5)​
By hypothests ar4−2r3=576 and ar−a=9. Dividing the last equation into the first yields r−1r4−r3​=64 so r3=64 and r=4. Since ar−2=9 and r=4,2=3 and therefore S5​=−33(1−45)​=1023.