Problem

Write a recursive formula for each sequence.
\begin{tabular}{|l|l|}
\hline $5,20,80,320, \ldots$ & $7,4,1,-2, \ldots$ \\
$a_{1}=\square$ & $a_{1}=\square$ \\
$a_{n}=\square$ for $n \geq 2$ & $a_{n}=\square$ for $n \geq 2$ \\
\hline
\end{tabular}

Answer

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Answer

Final Answer: \(\boxed{\begin{tabular}{|l|l|}\hline 5,20,80,320, \ldots & 7,4,1,-2, \ldots \\a_{1}=5 & a_{1}=7 \\a_{n}=4a_{n-1} \text{ for } n \geq 2 & a_{n}=a_{n-1}-3 \text{ for } n \geq 2 \\\hline\end{tabular}}\)

Steps

Step 1 :Identify the type of sequence. The first sequence appears to be a geometric sequence, where each term is multiplied by a constant to get the next term. The second sequence appears to be an arithmetic sequence, where a constant is subtracted from each term to get the next term.

Step 2 :Determine the ratio for the first sequence and the difference for the second sequence. The ratio for the first sequence is 4, which means each term is multiplied by 4 to get the next term. The difference for the second sequence is -3, which means 3 is subtracted from each term to get the next term.

Step 3 :Write the recursive formula for each sequence. For the first sequence, \(a_{1}=5\) and \(a_{n}=4a_{n-1}\) for \(n \geq 2\). For the second sequence, \(a_{1}=7\) and \(a_{n}=a_{n-1}-3\) for \(n \geq 2\).

Step 4 :Final Answer: \(\boxed{\begin{tabular}{|l|l|}\hline 5,20,80,320, \ldots & 7,4,1,-2, \ldots \\a_{1}=5 & a_{1}=7 \\a_{n}=4a_{n-1} \text{ for } n \geq 2 & a_{n}=a_{n-1}-3 \text{ for } n \geq 2 \\\hline\end{tabular}}\)

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