Chapter 11.2, Problem 5WE

To determine

To calculate:

The formula for $n^{th}$ terms of the following arithmetic sequence:

  $7,11,15,\mathrm{19...}$

Answer to Problem 5WE

The formula for $n^{th}$ terms of arithmetic sequence $7,11,15,\mathrm{19...}$ is $a_n=4n-3$ .

Explanation of Solution

Given information:

  $7,11,15,\mathrm{19...}$

Formula used:

The $n^{th}$ term of an arithmetic sequence is given by the following expression:

  $a_n=a_1+\left(n-1\right)d$

Where,

First term: $a_1$

Common difference: $d$

Calculation:

Consider the sequence as:

  $\begin{array}{l}{a}_1=7\\ a_2=11\\ a_3=15\\ a_4=19\end{array}$

Now, the common ratio is calculated as:

  $\begin{array}{l}d=a_2-a_1=11-7=4\\ d=a_3-a_2=16-11=4\\ d=a_4-a_3=19-15=4\end{array}$

Know that:

First term: $a_1=7$

Common difference: $d=4$

So, the $n^{th}$ term for given arithmetic sequence is derived as:

  $\begin{array}{l}{a}_n=a_1+\left(n-1\right)d\\ a_n=7+\left(n-1\right)\left(4\right)\\ a_n=7+4n-4\\ a_n=4n-3\end{array}$