Question

algebra 1
topic 3: linear functions review
write an explicit formula for the following recursive formula.
$a_n = a_{n - 1} + 13; a_1 = 8$
$a_n = \square$ (simplify your answer.)

Explanation:

Step1: Identify Sequence Type

The recursive formula \( a_n = a_{n - 1}+13 \) with \( a_1 = 8 \) is an arithmetic sequence, where the common difference \( d = 13 \) and the first term \( a_1 = 8 \).

Step2: Use Arithmetic Sequence Formula

The explicit formula for an arithmetic sequence is \( a_n = a_1+(n - 1)d \). Substitute \( a_1 = 8 \) and \( d = 13 \):
\( a_n = 8+(n - 1)\times13 \)

Step3: Simplify the Expression

Expand \( (n - 1)\times13 \): \( 13n-13 \). Then add 8:
\( a_n = 13n-13 + 8=13n - 5 \)

Answer:

\( 13n - 5 \)