Question

Consider the following geometric sequence 2, 6, 18, 54... what is the recursive formula? What is the explicit?

Answer 1

`a_1=2,\ a_2=6,\ a_3=18,\ a_4=54,\ ...`

The Geometric Sequence Formula

Explicit formula:
`a_n=a_1r^(n-1)`

Calculate the common ratio (r):

`r=(a_(n+1))/(a_n)\to r=(6)/(2)=3`

Substitute:

`a_n=2(3)^(n-1)=2\cdot3^n\cdot3^(-1)=2\cdot3^n\cdot(1)/(3)=(2)/(3)(3)^n`

Recursive formula:

`\left\{\begin{array}{ccc}a_1=2\\a_(n+1)=3a_n\end{array}\right`