Chapter 3, Problem 21PT

To determine

Whether the sequence $5,17,29,41,\dots$ is an arithmetic sequence, if yes, find $n\text{th}$ term of the sequence.

Answer to Problem 21PT

Yes, the sequence $5,17,29,41,\dots$ is an arithmetic sequence and $n\text{th}$ term is $a_n=12n-7$ .

Explanation of Solution

Given:

The sequence is $5,17,29,41,\dots$ .

Calculation:

Subtract the second and first term of the sequence to obtain the difference.

  $\begin{array}{c}\text{Difference}=17-5\\ =12\end{array}$

Subtract the fourth and third term of sequence to obtain the difference.

  $\begin{array}{c}\text{Difference}=29-17\\ =12\end{array}$

The difference is same; therefore, the sequence is an AP.

The first term is $5$ and the common difference is $12$ .

The $n\text{th}$ term of the sequence $5,17,29,41,\dots$ is calculated as:

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ =5+\left(n-1\right)\left(12\right)\\ =5+12n-12\\ =12n-7\end{array}$

Thus, the sequence $5,17,29,41,\dots$ is an arithmetic sequence and $n\text{th}$ term is $a_n=12n-7$ .