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Which recursive formula can be used to define this sequence for $n > 1$ ? $\newline$ $5, 19, 33, 47, 61, 75, \ldots$ $\newline$ Choices: $\newline$ (A) $a_n = a_{n-1} + 14$ $\newline$ (B) $a_n = a_{n-1} + a_{n-1} - 14$ $\newline$ (C) $a_n = 14a_{n-1}$ $\newline$ (D) $a_n = \frac{19}{5}a_{n-1}$

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Write a formula for a recursive sequence

Full solution

Q. Which recursive formula can be used to define this sequence for

n > 1

?

\newline

5, 19, 33, 47, 61, 75, \ldots

\newline

Choices:

\newline

(A)

a_n = a_{n-1} + 14

\newline

(B)

a_n = a_{n-1} + a_{n-1} - 14

\newline

(C)

a_n = 14a_{n-1}

\newline

(D)

a_n = \frac{19}{5}a_{n-1}

Determine Sequence Type: Analyze the sequence to determine if it is arithmetic or geometric. $\newline$ The sequence is $5, 19, 33, 47, 61, 75, \ldots$ $\newline$ To determine if it is arithmetic, we check if the difference between consecutive terms is constant. $\newline$ $19 - 5 = 14$ , $33 - 19 = 14$ , $47 - 33 = 14$ , and so on. $\newline$ Since the difference is constant, the sequence is arithmetic.
Find Common Difference: Find the common difference, $d$ , for the arithmetic sequence. $\newline$ Using two consecutive terms, $19$ and $5$ , we calculate the common difference. $\newline$ $d = 19 - 5 = 14$ $\newline$ The common difference ( $d$ ) is $14$ .
Identify Recursive Formula: Identify the recursive formula for the given arithmetic sequence. $\newline$ Since the sequence is arithmetic with a common difference of $14$ , the recursive formula will be of the form: $\newline$ $a_n = a_{n-1} + d$ $\newline$ Substituting the common difference, we get: $\newline$ $a_n = a_{n-1} + 14$

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