Questions: Complete the recursive rule and an explicit rule for the arithmetic sequence described by the table. Month n 1 2 3 4 5 Total Deposits () f(n) 89 102 115 128 141 f(1)= The common difference is . The recursive is f(1)= f(n)=f(n-1)+ for n ≥ . The explicit rule is f(n)= + (n-1)

Complete the recursive rule and an explicit rule for the arithmetic sequence described by the table.
Month n 1 2 3 4 5
Total Deposits () f(n) 89 102 115 128 141

f(1)=

The common difference is .

The recursive is f(1)= f(n)=f(n-1)+ for n ≥ .
The explicit rule is f(n)= + (n-1)

Transcript text: Complete the recursive rule and an explicit rule for the arithmetic sequence described by the table. \begin{tabular}{|l|c|c|c|c|c|c|} \hline Month & $n$ & 1 & 2 & 3 & 4 & 5 \\ \hline Total Deposits (\$) & $f(n)$ & 89 & 102 & 115 & 128 & 141 \\ \hline \end{tabular} \[ f(1)= \] $\square$ The common difference is $\square$ . The recursive is $f(1)=$ $\square$ $f(n)=f(n-1)+$ $\square$ for $n \geq$ $\square$ . The explicit rule is $f(n)=$ $\square$ $+$ $\square$ $(n-1)$

Solution

Solution Steps

To solve this problem, we need to identify the pattern in the arithmetic sequence given by the table. First, determine the common difference by subtracting consecutive terms. Then, use the first term and the common difference to complete the recursive and explicit formulas for the sequence.

Solution Approach

Identify the first term of the sequence, $ f(1) $.
Calculate the common difference by subtracting the first term from the second term.
Use the first term and the common difference to write the recursive formula.
Use the first term and the common difference to write the explicit formula.

Step 1: Identify the First Term

The first term of the arithmetic sequence is given by $ f(1) = 89 $.

Step 2: Calculate the Common Difference

The common difference $ d $ is calculated as follows: \[ d = f(2) - f(1) = 102 - 89 = 13 \]

Step 3: Write the Recursive Rule

The recursive rule for the sequence can be expressed as: \[ f(1) = 89, \quad f(n) = f(n-1) + 13 \quad \text{for } n \geq 2 \]

Step 4: Write the Explicit Rule

The explicit rule for the sequence is given by: \[ f(n) = 89 + 13(n - 1) \]