Chapter 12, Problem 20RE

To determine

To find:The general formula for the given arithmetic sequence.

Answer to Problem 20RE

The general formula for the given arithmetic sequenceis $a_n=5n-4$ .

Explanation of Solution

Given information:

The seventh and twenty terms are 31 and 96.

Calculation:

The $n^{th}$ term of arithmetic sequence is,

  $a_n=a_1+\left(n-1\right)d$.......(1)

The seventh term is,

  $\begin{array}{l}{a}_7=a_1+\left(7-1\right)d\\ 31=a_1+6d\end{array}$

The twentieth term is,

  $\begin{array}{l}{a}_{20}=a_1+\left(20-1\right)d\\ 96=a_1+19d\end{array}$

From above equation $d=5$ .

Substitute the value in above equation.

  $\begin{array}{l}31=a_1+6\left(5\right)\\ a_1=1\end{array}$

Substitute the values in Equation (1).

  $\begin{array}{l}{a}_n=1+\left(n-1\right)5\\ a_n=1+5n-5\\ a_n=5n-4\end{array}$