Question

What is the common ratio of the geometric sequence whose second and fourth terms are 6 and 54, respectively?

Answer 1

if a = first term and r = common ratio the second term = ar and 4th term = ar^3
So  54 / 6 = ar^3 / ar
9 = r^3 
r = 3
Commin ratio = 3

Answer 2

Since the nth term of any geometric sequence is:

a(n)=ar^(n-1), a(n)=nth term, a=initial term, r=common ratio, n=term number 

We can set up a ratio given the points as:

54/6=(ar^3)/(ar^1)

9=r^2

r=±3  however since the 4th term is greater than the second term we know r>0 so

r=3

The common ratio is 3.