Question

The sum of the first three terms of a geometric sequence of positive integers is equal to seven times the first term, and the sum of the first four terms is 45. What is the first term of the sequence

Answer 1

The first 3 terms are a, ar and ar^2
so a + ar + ar^2 = 7a

a + ar + ar^2 + ar^3 = 45
7a + ar^3 = 45
a(7 + r^3) = 45
a = 45/(7 + r^3)
Because we know the sequence are positive integers r must be a natural number.
(7 + r^3) is a factor of 45 [ie 1, 3, 5, 9, 15, 45]
so r = 2 because 7 + 2^3 = 15 is the only factor possible
a = 45/15 = 3