Which recursive sequence would produce the sequence $7,33,163, \ldots$ ? $\newline$ $a_{1}=7$ and $a_{n}=5 a_{n-1}-2$ $\newline$ $a_{1}=7$ and $a_{n}=5 a_{n-1}+4$ $\newline$ $a_{1}=7$ and $a_{n}=4 a_{n-1}+5$ $\newline$ $a_{1}=7$ and $a_{n}=-2 a_{n-1}+5$

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Divide numbers written in scientific notation

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Q. Which recursive sequence would produce the sequence

7,33,163, \ldots

\newline

a_{1}=7

and

a_{n}=5 a_{n-1}-2

\newline

a_{1}=7

and

a_{n}=5 a_{n-1}+4

\newline

a_{1}=7

and

a_{n}=4 a_{n-1}+5

\newline

a_{1}=7

and

a_{n}=-2 a_{n-1}+5

Identify First Term: Identify the first term of the sequence. $\newline$ The first term given is $a_{1} = 7$ .
Find Second Term: Use the first recursive formula to find the second term. $\newline$ The first option is $a_{n} = 5a_{n-1} - 2$ . Let's apply it to find $a_{2}$ . $\newline$ $a_{2} = 5 \times a_{1} - 2 = 5 \times 7 - 2 = 35 - 2 = 33$ . $\newline$ This matches the second term of the sequence.
Find Third Term: Use the first recursive formula to find the third term. $\newline$ Now, let's find $a_{3}$ using the same formula. $\newline$ $a_{3} = 5 \times a_{2} - 2 = 5 \times 33 - 2 = 165 - 2 = 163$ . $\newline$ This matches the third term of the sequence.
Verify Other Options: Verify that the other options do not match the sequence. $\newline$ To ensure we have the correct formula, we should check that the other options do not produce the sequence. $\newline$ For the second option, $a_{n} = 5a_{n-1} + 4$ , let's find $a_{2}$ . $\newline$ $a_{2} = 5 \times a_{1} + 4 = 5 \times 7 + 4 = 35 + 4 = 39$ . $\newline$ This does not match the second term of the sequence.
Check Second Option: Check the third option. $\newline$ For the third option, $a_{n} = 4a_{n-1} + 5$ , let's find $a_{2}$ . $\newline$ $a_{2} = 4 \times a_{1} + 5 = 4 \times 7 + 5 = 28 + 5 = 33$ . $\newline$ This matches the second term of the sequence, so we need to check the third term.
Check Third Option: Find the third term using the third option. $\newline$ $a_{3} = 4 \times a_{2} + 5 = 4 \times 33 + 5 = 132 + 5 = 137$ . $\newline$ This does not match the third term of the sequence.
Check Fourth Option: Check the fourth option. $\newline$ For the fourth option, $a_{n} = -2a_{n-1} + 5$ , let's find $a_{2}$ . $\newline$ $a_{2} = -2 \times a_{1} + 5 = -2 \times 7 + 5 = -14 + 5 = -9$ . $\newline$ This does not match the second term of the sequence.
Conclude Correct Formula: Conclude the correct recursive formula. $\newline$ Since only the first option produces the correct second and third terms of the sequence, we can conclude that the correct recursive formula is: $\newline$ $a_{1} = 7$ and $a_{n} = 5a_{n-1} - 2$ .