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Use the explicit formula to find a recursive formula for the sequence $a_n$ . Write your answer in simplest form. $\newline$ The recursive formula should depend on $a_{n - 1}$ . $\newline$ $a_n = -2n + 29$ $\newline$ $a_n =$ ______

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Convert an explicit formula to a recursive formula

Full solution

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

a_n

. Write your answer in simplest form.

\newline

The recursive formula should depend on

a_{n - 1}

.

\newline

a_n = -2n + 29

\newline

a_n =

______

Identify Sequence Type: Identify if the given sequence is geometric or arithmetic. $\newline$ $a_n = -2n + 29$ does not involve any exponent terms, which means it is not geometric. Since the formula involves a constant difference between terms (indicated by the linear term $-2n$ ), the sequence is arithmetic.
Find First Term: Find the first term of the sequence using the explicit formula. $\newline$ $a_1 = -2(1) + 29$ $\newline$ $= -2 + 29$ $\newline$ $= 27$ $\newline$ This is the value of the sequence when $n = 1$ .
Find Second Term: Find the second term of the sequence using the explicit formula. $\newline$ $a_2 = -2(2) + 29$ $\newline$ $= -4 + 29$ $\newline$ $= 25$ $\newline$ This is the value of the sequence when $n = 2$ .
Find Common Difference: Find the common difference in the arithmetic sequence. $\newline$ $d = a_2 - a_1$ $\newline$ $= 25 - 27$ $\newline$ $= -2$ $\newline$ The common difference $d$ is the amount by which each term decreases to get to the next term.
Write Recursive Formula: Write the recursive formula by plugging in the value of the common difference. $\newline$ Substitute $-2$ for $d$ in $a_n = a_{(n - 1)} + d$ . $\newline$ $a_n = a_{(n - 1)} - 2$ $\newline$ This recursive formula expresses each term of the sequence as the previous term minus $2$ .

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