Chapter 12, Problem 40RE

To determine

To find: The general form for the arithmetic sequence.

Answer to Problem 40RE

The general from of the sequence is,

  $a_n=2n+6$

Explanation of Solution

Given:

The 12th term is $30$ and the 22nd term is $50$ .

Calculation:

Consider the general term of the arithmetic sequence is,

  $a_n=a_1+\left(n-1\right)d$

For 22nd term is $50$ .

  $\begin{array}{l}50=a_1+\left(22-1\right)d\\ 50=a_1+21d\end{array}$............(I)

For the 12th term is $30$

  $\begin{array}{l}30=a_1+\left(12-1\right)d\\ 30=a_1+11d\end{array}$

From equation (I) and from above,

  $\begin{array}{l}50-30=a_1+21d-a_1+11d\\ 10d=20\\ d=2\end{array}$

Then,

  $\begin{array}{l}50=a_1+21\left(2\right)\\ a_1=8\end{array}$

Then, the general form of the sequence is,

  $\begin{array}{c}{a}_n=8+\left(n-1\right)\left(2\right)\\ =2n+6\end{array}$