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Use the initial term and the recursive formula to find an explicit formula for the sequence $a_n$ . Write your answer in simplest form. $\newline$ $a_1 = 1$ $\newline$ $a_n = a_{n - 1} + 14$ $\newline$ $a_n =$ ______

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Convert a recursive formula to an explicit formula

Full solution

Q. Use the initial term and the recursive formula to find an explicit formula for the sequence

a_n

. Write your answer in simplest form.

\newline

a_1 = 1

\newline

a_n = a_{n - 1} + 14

\newline

a_n =

______

Given Sequence Type: We have: $\newline$ Initial term $a_1 = 1$ $\newline$ Recursive formula: $a_n = a_{n-1} + 14$ $\newline$ Is the given sequence geometric or arithmetic? $\newline$ The recursive formula $a_n = a_{n-1} + 14$ corresponds to $a_n = a_{n - 1} + d$ . $\newline$ Hence, the given sequence is arithmetic.
Common Difference: Recursive formula: $a_n = a_{n-1} + 14$ $\newline$ Find the common difference in the arithmetic sequence. $\newline$ Compare $a_n = a_{n-1} + 14$ with $a_n = a_{n - 1} + d$ to find $d$ . $\newline$ So, $d = 14$ .
Explicit Formula Calculation: Explicit formula for arithmetic sequence: $\newline$ $a_n = a_1 + d(n - 1)$ $\newline$ Here, $a_1$ is the first term, and $d$ is the common difference. We have: $\newline$ $a_1 = 1$ $\newline$ $d = 14$ $\newline$ Find the explicit formula. $\newline$ Substitute $1$ for $a_1$ and $14$ for $d$ in $a_n = a_1 + d(n - 1)$ . $\newline$ $a_1$ $0$ $\newline$ $a_1$ $1$ $\newline$ $a_1$ $2$ $\newline$ $a_1$ $3$ $\newline$ Explicit formula is: $a_1$ $4$

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