Chapter 9.2, Problem 27E

To determine

To Find: The formula for $a_n$ for the arithmetic sequence.

Answer to Problem 27E

The $n^{th}$ term of the sequence is $a_n=\frac{5}{3}+\frac{10}{3}n$ .

Explanation of Solution

Given information:

  $a_1=5\&a_4=15$

Concept and Formula Used:

A sequence is airthmetic when the differences between consecutive terms are the same.

General term of the sequence: $a_n=a+\left(n-1\right)d$

where $a$ is the first term and $d$ is the common difference.

Calculation:

It is given that $a_1=a=5\&a_4=15$

Using the formula for $n^{th}$ term

  $\begin{array}{l}{a}_n=a+\left(n-1\right)d\\ a_4=5+\left(4-1\right)d\\ 15=5+3d\\ 10=3d\\ d=\frac{10}{3}\end{array}$

Now, for $n^{th}$ term

  $\begin{array}{l}{a}_n=a+\left(n-1\right)d\\ a_n=5+\left(n-1\right)\frac{10}{3}\\ a_n=5+\frac{10}{3}n-\frac{10}{3}\\ a_n=\frac{5}{3}+\frac{10}{3}n\end{array}$

Conclusion:

The $n^{th}$ term of the sequence is $a_n=\frac{5}{3}+\frac{10}{3}n$ .