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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 $a_{n - 1}$ . $\newline$ $a_n = 5n + 24$ $\newline$ $a_n = \_$

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Convert between explicit and recursive formulas

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 = 5n + 24

\newline

a_n = \_

Write Sequence Terms: Write down the first few terms of the sequence by substituting $n = 1, 2, 3, 4$ into the explicit formula $a_n = 5n + 24$ . $a_1 = 5(1) + 24 = 29$ $a_2 = 5(2) + 24 = 34$ $a_3 = 5(3) + 24 = 39$ $a_4 = 5(4) + 24 = 44$
Identify Common Difference: Notice the common difference between consecutive terms. $\newline$ Difference between $a_2$ and $a_1 = 34 - 29 = 5$ $\newline$ Difference between $a_3$ and $a_2 = 39 - 34 = 5$ $\newline$ Difference between $a_4$ and $a_3 = 44 - 39 = 5$
Determine Recursive Formula: Since the common difference is $5$ , the sequence is arithmetic and the recursive formula is $a_n = a_{n - 1} + d$ , where $d$ is the common difference.
Write Recursive Formula: Write the recursive formula using the common difference found in Step $2$ . $\newline$ $a_n = a_{n - 1} + 5$

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