Question

PLEASE HELP, TIMED FINAL

A sequence is defined by the explicit formula an = 3^n + 4. Which recursive formula represents the same sequence of numbers?

Answer 1

`a_(n)` = 3
`a_(n-1)` - 8, with
`a_(1)` = 7

generate the first few terms of the sequence

`a_(1)` = 3 + 4 = 7

`a_(2)` = 3² + 4 = 9 + 4 = 13

`a_(3)` = 3³ + 4 = 237 + 4 = 31

`a_(4)` =
`3^(4)` + 4 = 81 + 4 = 85

the sequence is 7, 13, 31, 85, .....

Checking the recursive formulae given the one that generates the sequence is

`a_(n)` = 3
`a_(n-1)` - 8 with
`a_(1)` = 7, as

`a_(2)` = (3 × 7 ) - 8 = 21 - 8 = 13 ← correct

`a_(3)` = (3 × 13 ) - 8 = 39 - 8 = 31 ← correct

`a_(4)` = (3 × 31 ) - 8 = 93 - 8 = 85 ← correct

Answer 2

We'll analyze all options:

`\bold{a)}~a_n =3a_(n-2) +4\\\\a_n = 3\cdot(3^(n-2)+4)+4\\\\a_n = 3^(n-1) + 12 + 4\\\\a_n = 3^(n-1) + 18\Longrightarrow False!\\\\\\\bold{b)}~a_n = 3n+a_(n-1)\\\\a_n = 3n + 3^(n-1)+4\Longrightarrow False!\\\\\\\bold{c)}~a_n = 3a_(n-1) -8\\\\a_n = 3\cdot(3^(n-1)+4)-8\\\\a_n = 3^n +12-8\\\\\boxed{a_n = 3^n +4}\Longrightarrow True!\\\\\\\bold{d)}~False!`

The first term is
`a_1 = 3^1+4=7`, what is correct for all options.

Then, the correct answer is the third (C).