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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?

PLEASE HELP TIMED FINAL A sequence is defined by the explicit formula an 3n 4 Which recursive formula represents the same sequence of numbers class=

Respuesta :

arthurpdc

We'll analyze all options:

[tex]\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![/tex]

The first term is [tex]a_1 = 3^1+4=7[/tex], what is correct for all options.

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

[tex]a_{n}[/tex] = 3[tex]a_{n-1}[/tex] - 8, with [tex]a_{1}[/tex] = 7

generate the first few terms of the sequence

[tex]a_{1}[/tex] = 3 + 4 = 7

[tex]a_{2}[/tex] = 3² + 4 = 9 + 4 = 13

[tex]a_{3}[/tex] = 3³ + 4 = 237 + 4 = 31

[tex]a_{4}[/tex] = [tex]3^{4}[/tex] + 4 = 81 + 4 = 85

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

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

[tex]a_{n}[/tex] = 3[tex]a_{n-1}[/tex] - 8 with [tex]a_{1}[/tex] = 7, as

[tex]a_{2}[/tex] = (3 × 7 ) - 8 = 21 - 8 = 13 ← correct

[tex]a_{3}[/tex] = (3 × 13 ) - 8 = 39 - 8 = 31 ← correct

[tex]a_{4}[/tex] = (3 × 31 ) - 8 = 93 - 8 = 85 ← correct








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