Question

1. what is the equivalent recursive definition for $a_n = 12 + (n - 1)3$?

$a_1 = 12, a_n = 3a_{n - 1}$
$a_1 = n, a_n = 12a_{n - 1} + 3$
$a_1 = 3, a_n = a_{n - 1} + 12$
$a_1 = 12, a_n = a_{n - 1} + 3$

Explanation:

Step1: Identify initial term

Substitute $n=1$ into $a_n = 12+(n-1)3$:
$a_1 = 12+(1-1)3 = 12$

Step2: Find common difference

The given formula is arithmetic sequence form $a_n = a_1+(n-1)d$, so $d=3$.

Step3: Recursive formula for arithmetic sequence

Recursive rule: $a_n = a_{n-1}+d$, so $a_n = a_{n-1}+3$.

Step4: Match with options

The recursive definition is $a_1=12$, $a_n=a_{n-1}+3$.

Answer:

$a_1 = 12,\ a_n = a_{n-1} + 3$