Geometric series

Question

Given A_n=(-4)(6)^n-1 for a geometric sequence, determine the common ratio:

Answer 1

The common ratio is the ratio of any two consecutive terms, A_n+1/A_n

A_n = (-4)·6^n-1

A_n+1 = (-4)·6ⁿ

A_n+1/A_n = (-4)·6ⁿ / (-4)·6^n-1 = 6

The common ratio is 6. Note that this is the base of the exponent term 6^n-1. The explicit form of a geometric series is:

A_n = A₁·r^n-1

where A₁ is the first term in the series and r is the common ratio. So in your equation A₁ = -4 and r = 6.