geometric sequence

Question

For a given geometric sequence, the 6th term, a6, is equal to 13/16, and the 9th term, a9 is equal to 52. Find the value of the 12th term ,a12. If applicable write your answer as a fraction.

Answer 1

The nth term of a geometric sequence is a[0]x^n, so the 6th term is a[0]x^6 and the 9th term is a[0]x^9. If we divide these two terms we get x^3, so x^3=52 divided by 13/16, which is 52*16/13=64 and x=4. [To find a[0] we take the 6th or 9th term and use its value and the value of x: let's take the 9th term, 52. x^9=4^9=262144, so 262144*a[0]=52, making a[0]=52/262144=13/65536. 4^6*13/65536=13/16, which is the 6th term. The 12th term is 4^12*13/65536=3328.]

An alternative method is to multiply the 9th term by x^3 (x^9*x^3=x^12). So 52*64=3328. This method doesn't require us to find a[0].