Question

write an explicit formula that represents the sequence defined by following recursive formula:
$a_1 = 3$ and $a_n = a_{n-1} + 7$
answer attempt 1 out of 3
$a_n = \square$

Explanation:

Step1: Identify Sequence Type

The recursive formula \( a_1 = 3 \) and \( a_n = a_{n - 1}+7 \) shows a common difference (\( d = 7 \)) between consecutive terms, so it's an arithmetic sequence.

Step2: Recall Arithmetic Sequence Formula

The explicit formula for an arithmetic sequence is \( a_n=a_1+(n - 1)d \), where \( a_1 \) is the first term and \( d \) is the common difference.

Step3: Substitute Values

Here, \( a_1 = 3 \) and \( d = 7 \). Substitute into the formula:
\( a_n=3+(n - 1)\times7 \)
Simplify: \( a_n=3 + 7n-7=7n - 4 \).

Answer:

\( 7n - 4 \)