Chapter 8.2, Problem 37E

To determine

To find:

The $n\text{th}$ term of the sequence.

Answer to Problem 37E

The $n\text{th}$ term of an Arithmetic sequence is $24-3n$ .

Explanation of Solution

Given information:

The arithmetic sequence whose first $10$ terms are represented by the graph.

  

From the given figure it can be observed that $a_4=12$ and $a_5=9$ .

The $n\text{th}$ term of an Arithmetic sequence is:

  $a_n=a_1+\left(n-1\right)d$

Substitute $a_4=12$ and $n=4$ in the above formula.

  $\begin{array}{c}12=a_1+\left(4-1\right)d\\ =a_1+3d\mathrm{...........}\left(1\right)\end{array}$

Substitute $a_5=9$ and $n=5$ in the above formula.

  $\begin{array}{c}9=a_1+\left(5-1\right)d\\ =a_1+4d\mathrm{............}\left(2\right)\end{array}$

Subtract equation $\left(1\right)$ from equation $\left(2\right)$ .

  $\begin{array}{c}{a}_1+4d-\left(a_1+3d\right)=9-12\\ 4d-3d=-3\\ d=-3\end{array}$

Substitute $d=-3$ in the equation $\left(1\right)$ .

  $\begin{array}{c}{a}_1+3\left(-3\right)=12\\ a_1-9=12\\ a_1=12+9\\ a_1=21\end{array}$

The $n\text{th}$ term of an Arithmetic sequence is:

  $a_n=a_1+\left(n-1\right)d$

Substitute $a_1=21$ and $d=-3$ in the above formula:

  $\begin{array}{c}{a}_n=21+\left(n-1\right)\left(-3\right)\\ =21-3\left(n-1\right)\\ =21-3n+3\\ =24-3n\end{array}$

Hence, the $n\text{th}$ term of an Arithmetic sequence is $24-3n$ .