Question

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

Answer 1

Answer:

$Recursive\ formula\ a_n=3a_{n-1}-8,\ a_1=7$

Step-by-step explanation:

Recursive Formula: The formula that defines the every term of sequence using previous terms.

$a_n=3^n+4\\\\a_1=3^1+4\\\\a_1=3+4\\\\a_1=7$

$a_n=3^n+4...................................(1)\\\\a_{n-1}=3^{n-1}+4..........................(2)\\\\Multiply\ by\ 3\ both\ the\ sides\\\\3a_{n-1}=3\times (3^{n-1}+4)\\\\3a_{n-1}=3\times 3^{n-1}+3\times 4\\\\3a_{n-1}=3^{n}+12\\\\3a_{n-1}==3^n+4+8\\\\From\ eq(1)\ 3^n+4=a_n\\\\3a_{n-1}=a_n+8\\\\Subtract\ 8\ from\ the\ both\ sides\\\\3a_{n-1}-8=a_n+8-8\\\\a_n=3a_{n-1}-8$

$Recursive\ formula\ a_n=3a_{n-1}-8,\ a_1=7$