Question

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 the sequence type

The recursive formula is \( a_n = a_{n - 1}+13 \) with \( a_1 = 8 \). This is an arithmetic sequence because each term is obtained by adding a constant difference (13) to the previous term. The general form of an arithmetic sequence's explicit formula is \( a_n=a_1+(n - 1)d \), where \( a_1 \) is the first term and \( d \) is the common difference.

Step2: Substitute values into the formula

Here, \( a_1 = 8 \) and \( d = 13 \). Substituting these into the formula:
\( a_n=8+(n - 1)\times13 \)

Step3: Simplify the expression

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

Answer:

\( 13n - 5 \)