Question

Given the sequence rn defined recursively below, find r4.

r1=2

rn=4rn−1−3

Answer 1

`r_4 = 65`

Explanation:

Given

`r_1 = 2`

`r_n = 4r_(n-1) - 3`

Required

Find
`r_4`

Calculating the value of
`r_4`

This means n = 4;

Hence,

`r_n = 4r_(n-1) - 3`

`r_4 = 4r_(4-1) - 3`

`r_4 = 4r_3 - 3`

At this point, we need to solve for
`r_3`;

Taking n as 3

`r_n = 4r_(n-1) - 3`

`r_3 = 4r_(3-1) - 3`

`r_3 = 4r_2 - 3`

At this point, we need to solve for
`r_2`;

Taking n as 2

`r_n = 4r_(n-1) - 3`

`r_2 = 4r_(2-1) - 3`

`r_2 = 4r_1 - 3`

Substitute 2 for
`r_1`

`r_2 = 4 * 2 - 3`

`r_2 = 8 - 3`

`r_2 = 5`

Solving for
`r_3`

Substitute 5 for
`r_2`

`r_3 = 4 * 5 - 3`

`r_3 = 20 - 3`

`r_3 = 17`

Solving for
`r_4`

Substitute 17 for
`r_3`

`r_4 = 4 * 17 - 3`

`r_4 = 68 - 3`

`r_4 = 65`

Hence,
`r_4 = 65`