Chapter 4.6, Problem 55E

To determine

To write: The function that represents the arithmetic sequence in the mapping diagram

Answer to Problem 55E

The function that represents the arithmetic sequence in the mapping diagram is $f\left(n\right)=6n+17$ .

Explanation of Solution

Given:

Consider the arithmetic sequence shown in the mapping diagram

  

Calculation:

From the diagram terms of the arithmetic sequence are

  $f\left(1\right)=23,f\left(4\right)=39.5\text{ and }f\left(12\right)=89$

Consider the standard form for the term $f\left(n\right)=a_1+\left(n-1\right)d$ where $a_1$ is first term and d is the difference.

Now consider the 12th term

  $\begin{array}{l}f\left(12\right)=23+\left(12-1\right)d\hfill \\ 89=23+11d\hfill \end{array}$

Take the variable on one side and find the value

  $11d=66$

Divide by 11 on both sides

  $d=6$

So, the function for $n^{th}$ term is

  $\begin{array}{l}f\left(n\right)=a_1+\left(n-1\right)d\hfill \\ =23+\left(n-1\right)\left(6\right)\hfill \\ =23+\text{6}n-6\hfill \\ =6n+17\hfill \end{array}$

Conclusion:

Therefore, the function that represents the arithmetic sequence in the mapping diagram is $f\left(n\right)=6n+17$ .