Chapter 11.4, Problem 59SR

To determine

Find the missing value of $a_8$ in arithmetic sequence.

Answer to Problem 59SR

  $\boxed{a_8=-57}$

Explanation of Solution

Given information:

Find the missing value for each arithmetic sequence.

  $a_8=?,a_{19}=31,d=8$

Calculation:

Consider the standard $n^{th}$ term of arithmetic sequence is,

  $a_n=a_1+\left(n-1\right)d$

Where $a$ the first is term and $d$ is the common difference between each consecutive terms.

  $a_8=?,a_{19}=31,d=8$

Put the values in $a_n=a_1+\left(n-1\right)d$ equation,

  $\begin{array}{l}{a}_{19}=a_1+\left(19-1\right)7\\ 31=a_1+144\\ a_1=-113\end{array}$

Now substitute $a_1=-113,d=8,n=8$ , in standard equation,

  $\begin{array}{l}{a}_8=-113+\left(8-1\right)8\\ a_8=-113+56\\ a_8=-57\end{array}$

Hence, the value of $a_8$ is $\boxed{a_8=-57}$ .