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Use the explicit formula to find a recursive formula for the sequence $a$ . Write your answer in simplest form. $\newline$ The recursive formula should depend on $a$ . $\newline$ $a = n - 9$ $\newline$ a = ______

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Convert an explicit formula to a recursive formula

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Q. Use the explicit formula to find a recursive formula for the sequence

a

. Write your answer in simplest form.

\newline

The recursive formula should depend on

a

.

\newline

a = n - 9

\newline

a = ______

Write Formulas: To find a recursive formula for the sequence, we need to express the nth term $a_n$ in terms of the previous term $a_{(n-1)}$ . The explicit formula given is $a = n - 9$ . Let's start by writing the formula for the nth term and the $(n-1)$ th term. $\newline$ $a_n = n - 9$ $\newline$ $a_{(n-1)} = (n-1) - 9$
Express in Terms: Now, we will express $a_n$ in terms of $a_{(n-1)}$ . To do this, we can add $1$ to both sides of the equation for $a_{(n-1)}$ to get the equation for $a_n$ . $\newline$ $a_{(n-1)} + 1 = (n-1) - 9 + 1$ $\newline$ $a_{(n-1)} + 1 = n - 9$ $\newline$ Notice that the right side of this equation is the same as the explicit formula for $a_n$ . Therefore, we can write: $\newline$ $a_n = a_{(n-1)} + 1$
Find Recursive Formula: We have now found a recursive formula for the sequence. The recursive formula is: $a_n = a_{n-1} + 1$ , where $a_1 = 1 - 9 = -8$ (since the first term of the sequence when $n=1$ is $-8$ ).

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