Chapter 11.4, Problem 61SR

To determine

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

Answer to Problem 61SR

  $\boxed{n=20}$

Explanation of Solution

Given information:

Find the missing value for each arithmetic sequence.

  $a_7=-28,a_n=76,d=8,n=?$

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.

  $a_7=-28,a_n=76,d=8,n=?$

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

  $\begin{array}{l}{a}_7=a_1+\left(7-1\right)8\\ -28=a_1+48\\ a_1=-76\end{array}$

Now substitute $a_n=76,d=8,a_1=-76$ , in standard equation,

  $\begin{array}{l}76=-76+\left(n-1\right)8\\ 76=-76+8n-8\\ 76=-84+8n\\ 8n=160\\ n=\frac{160}{8}\\ n=20\end{array}$

Hence, the value of $n$ is $\boxed{n=20}$ .