Question

find the explicit rule for the nth term of the geometric sequence if the second and fifth terms in the sequence are 12 ans 324, respectively.

Answer 1

`a_(n) = 4*3^(n-1)`

Explanation:

Step-by-step explanation:

Geometric sequence concepts:

The nth term of a geometric sequence is given by the following equation.

`a_(n+1) = ra_(n)`

In which r is the common ratio.

This can be expanded for the nth term in the following way:

`a_(n) = a_(1)r^(n-1)`

Or also

`a_(n) = a_(m)r^(n-m)`

In which
`a_(1)` is the first term

In this question:

`a_(2) = 12, a_(5) = 324`

Then

Finding the common ratio:

`a_(n) = a_(m)r^(n-m)`

`a_(5) = a_(2)r^(5-2)`

`12r^(3) = 324`

`r^(3) = (324)/(12)`

`r = \sqrt[3]{(324)/(12)}`

`r = 3`

Finding the first term:

`a_(n) = a_(1)r^(n-1)`

`a_(2) = a_(1)*r`

`a_(1) = (a_(2))/(r) = (12)/(3) = 4`

Explicit rule:

`a_(n) = a_(1)r^(n-1)`

`a_(n) = 4*3^(n-1)`