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Which recursive formula can be used to define this sequence for $n > 1$ ? $\newline$ $5, 18, 31, 44, 57, 70, \ldots$ $\newline$ Choices: $\newline$ (A) $a_n = a_{n-1} + a_{n-2} - 13$ $\newline$ (B) $a_n = a_{n-1} + 13$ $\newline$ (C) $a_n = a_{n-1} - 13$ $\newline$ (D) $a_n = \frac{18}{5}a_{n-1}$

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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, 18, 31, 44, 57, 70, \ldots

\newline

Choices:

\newline

(A)

a_n = a_{n-1} + a_{n-2} - 13

\newline

(B)

a_n = a_{n-1} + 13

\newline

(C)

a_n = a_{n-1} - 13

\newline

(D)

a_n = \frac{18}{5}a_{n-1}

Analyze Sequence: Analyze the sequence to determine if it is arithmetic or geometric. $\newline$ The sequence is $5, 18, 31, 44, 57, 70, \ldots$ . To determine if it is arithmetic or geometric, we can look at the differences or ratios between terms. Let's calculate the difference between the first two terms: $18 - 5 = 13$ . Now, let's check the difference between the second and third terms: $31 - 18 = 13$ . Since the difference is consistent, we can conclude that the sequence is arithmetic with a common difference of $13$ .
Identify Recursive Formula: Identify the recursive formula for the given arithmetic sequence. $\newline$ Since we have determined that the sequence is arithmetic with a common difference of $13$ , the recursive formula will involve adding $13$ to the previous term to get the next term. Therefore, the recursive formula will be of the form $a_n = a_{n-1} + d$ , where $d$ is the common difference.
Substitute Common Difference: Substitute the common difference into the recursive formula. $\newline$ We have found that the common difference $d$ is $13$ . Substituting this into the recursive formula gives us $a_n = a_{(n-1)} + 13$ .
Match Recursive Formula: Match the correct recursive formula with the given choices. $\newline$ The recursive formula we have found is $a_n = a_{n-1} + 13$ . Now, let's look at the choices provided: $\newline$ (A) $a = a + a - 13$ (This is not a valid recursive formula; it doesn't make sense mathematically.) $\newline$ (B) $a = a + 13$ (This choice is missing the subscript $n$ and $(n-1)$ to denote the terms in the sequence.) $\newline$ (C) $a = a - 13$ (This would imply the sequence is decreasing by $13$ , which is not the case.) $\newline$ (D) $a = \frac{18}{5}a$ (This implies a geometric sequence with a ratio of $\frac{18}{5}$ , which is not correct.) $\newline$ The correct choice that matches our recursive formula is not listed exactly as we have it, but choice (B) is the closest if we correct it to include the proper subscripts: $a_n = a_{n-1} + 13$ .

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