Question

construct an explicit rule in function notation for the arithmetic sequence re determine the value of the given term, and explain what it means. darnell starts saving the same amount from each week’s paycheck. the table f(n) of his savings account over time in weeks. time (weeks) n: 1, 2, 3, 4; savings account balance ($) f(n): 310, 350, 390, 430. determine the value of f(12) and tell what it represents in this situation. f(n)=310+ (n - 1) f(12)= . after 12 weeks, darnell’s savings account will have a total of $ .

Explanation:

Step1: Find the common difference

The common difference $d$ of an arithmetic - sequence can be found by subtracting consecutive terms. $d = 350 - 310=40$.

Step2: Write the explicit formula

The explicit formula for an arithmetic sequence in function notation is $f(n)=a_1 + d(n - 1)$, where $a_1$ is the first - term and $d$ is the common difference. Here, $a_1 = 310$ and $d = 40$, so $f(n)=310+40(n - 1)$.

Step3: Calculate $f(12)$

Substitute $n = 12$ into the formula:
\[

$$\begin{align*} f(12)&=310+40(12 - 1)\\ &=310+40\times11\\ &=310 + 440\\ &=750 \end{align*}$$

\]

Answer:

$f(n)=310 + 40(n - 1)$; $f(12)=750$. After 12 weeks, Darnell's savings account will have a total of $\$750$.