Question

The sum of the first 3 terms of a G.P. is 27 and the sum of the fourth, fifth and sixth terms is -1. Find the common ratio and the sum to infinity of the G.P.

Answer 1

a + ar + ar^2 = 27
a(1 + r + r^2) = 27

the sum of the fourth, fifth and sixth terms is -1
---> ar^3 + ar^4 + ar^5 = -1
ar^3(1 + r + r^2) = -1

divide the 2nd by the 1st, the a would cancel and the (1 + r + r^2)
would cancel leaving

r^3 = -1/27
r = -1/3

back in a(1 + r + r^2) = 27
a(1 - 1/3 + 1/9) = 27
a(7/9) = 27
a = 27*9/7 = 243/7

sum(all terms) = a/(1-r)
= (243/7) / (4/3) = 729/28