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