Question

8-122. an arithmetic sequence has f(8) = 1058 and f(13) = 116. write an equation for the sequence. what is f(5)?

Explanation:

Step1: Recall arithmetic sequence formula

The nth term of an arithmetic sequence is given by $f(n) = f(1) + (n-1)d$, where $f(1)$ is the first term and $d$ is the common difference.

Step2: Set up equations for given terms

For $n=8$: $f(8) = f(1) + 7d = 106$
For $n=13$: $f(13) = f(1) + 12d = 116$

Step3: Solve for common difference $d$

Subtract the first equation from the second:
$$(f(1) + 12d) - (f(1) + 7d) = 116 - 106$$
$$5d = 10$$
$$d = 2$$

Step4: Solve for first term $f(1)$

Substitute $d=2$ into $f(1) + 7d = 106$:
$$f(1) + 7(2) = 106$$
$$f(1) + 14 = 106$$
$$f(1) = 106 - 14 = 92$$

Step5: Write the sequence equation

Substitute $f(1)=92$ and $d=2$ into the nth term formula:
$$f(n) = 92 + (n-1)(2)$$
Simplify the formula:
$$f(n) = 92 + 2n - 2 = 2n + 90$$

Answer:

The equation for the sequence is $f(n) = 2n + 90$ (or $f(n) = 92 + (n-1)2$), and the first term $f(1)=92$.