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Which recursive formula can be used to define this sequence for $n > 1$ ? $\newline$ $15, 28, 41, 54, 67, 80, \ldots$ $\newline$ Choices: $\newline$ (A) $a = \frac{27}{14}a$ $\newline$ (B) $a = \frac{1}{13}a$ $\newline$ (C) $a = a + 13$ $\newline$ (D) $a = a + a - 13$

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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

15, 28, 41, 54, 67, 80, \ldots

\newline

Choices:

\newline

(A)

a = \frac{27}{14}a

\newline

(B)

a = \frac{1}{13}a

\newline

(C)

a = a + 13

\newline

(D)

a = a + a - 13

Sequence Type: We have the sequence: $15, 28, 41, 54, 67, 80, \ldots$ $\newline$ Is the given sequence geometric or arithmetic? $\newline$ The difference between consecutive terms appears to be constant. $\newline$ The given sequence is likely arithmetic.
Calculate Common Difference: To confirm that the sequence is arithmetic, we calculate the difference between consecutive terms. $\newline$ The difference between the first two terms is $28 - 15 = 13$ . $\newline$ The difference between the second and third terms is $41 - 28 = 13$ . $\newline$ Since the difference is the same, the sequence is indeed arithmetic with a common difference of $13$ .
Identify Recursive Formula: Now, we need to identify the recursive formula for the given arithmetic sequence. $\newline$ The recursive formula for an arithmetic sequence is generally given by $a_n = a_{n-1} + d$ , where $d$ is the common difference. $\newline$ Substituting the common difference of $13$ into the formula, we get $a_n = a_{n-1} + 13$ .
Correct Recursive Formula: Looking at the given choices, we can see that the correct recursive formula that matches our calculation is (C) $a = a + 13$ . However, the choice should be written more precisely as $a_n = a_{n-1} + 13$ .

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