Question

A geometric sequence is defined by the equation an = (3)3 − n.

Part A: What are the first three terms of the sequence? (1 point)

Part B: What is the value of r? (2 points)

Part C: What is the value of a11? (2 points)

Answer 1

PART A

The geometric sequence is defined by the equation


`a_(n)=3^(3-n)`

To find the first three terms, we put n=1,2,3

When n=1,


`a_(1)=3^(3-1)`


`a_(1)=3^(2)`


`a_(1)=9`
When n=2,


`a_(2)=3^(3-2)`

`a_(2)=3^(1)`


`a_(2)=3`

When n=3


`a_(3)=3^(3-3)`


`a_(3)=3^(0)`

`a_(1)=1`
The first three terms are,


`9,3,1`

PART B

The common ratio can be found using any two consecutive terms.

The common ratio is given by,

`image`

`r = (3)/(9)`


`r = (1)/(3)`

PART C

To find

`a_(11)`

We substitute n=11 into the equation of the geometric sequence.


`a_(11) = {3}^(3 - 11)`

This implies that,


`a_(11) = {3}^( - 8)`


`a_(11) = \frac{1}{ {3}^(8) }`


`a_(11)=(1)/(6561)`