Question

a recursive formula is $a_1 = 11$, $a_n = a_{n - 1} - 2$. (simplify your answers.) an explicit formula is $a_n = \square$. (simplify your answer.)

Explanation:

Step1: Identify Sequence Type

The recursive formula \( a_1 = 11 \), \( a_n = a_{n - 1}-2 \) shows an arithmetic sequence with first term \( a_1 = 11 \) and common difference \( d=-2 \).

Step2: Recall Arithmetic Explicit Formula

The explicit formula for an arithmetic sequence is \( a_n = a_1+(n - 1)d \).

Step3: Substitute Values

Substitute \( a_1 = 11 \) and \( d=-2 \) into the formula:
\( a_n=11+(n - 1)(-2) \).

Step4: Simplify the Expression

Expand and simplify:
\( a_n=11-2n + 2=13-2n \).

Answer:

\( 13 - 2n \)