Chapter 11.2, Problem 78HP

To determine

Find an arithmetic series with 8 terms and a sum of 324.

Answer to Problem 78HP

  $a_n=\{16,23,30,37,44,51,58,\mathrm{65.....}\}$

Explanation of Solution

Given:

Write an arithmetic series with 8 terms and a sum of 324.

Concept Used:

We have to determine an arithmetic sequence $a_{\text{n}}$ with the given data: ${\text{S}}_{\text{n}}=324$

The formula for the sum of any arithmetic series is ${\text{S}}_{\text{n}}=\frac{n}{2}(a_1+a_{\text{n}})$

Calculation:

We have to determine an arithmetic sequence $a_{\text{n}}$ with the given data: ${\text{S}}_{\text{n}}=324$

The formula for the sum of any arithmetic series is ${\text{S}}_{\text{n}}=\frac{n}{2}(a_1+a_{\text{n}})$

  $\begin{array}{l}{\text{S}}_{\text{n}}=\frac{n}{2}(a_1+a_{\text{n}})\\ {\text{S}}_{\text{8}}=\frac{8}{2}(a_1+a_8)=324[\text{Substitute}n=8]\\ 4((a_1+a_8)=324\\ (a_1+a_{\text{8}})=\frac{324}{4}=81\\ a_1+a_{\text{8}}=81\end{array}$

Let`s give $a_1=16$ an arbitrary value and find the common difference d :

  $\begin{array}{l}{a}_1+a_{\text{8}}=81\Rightarrow a_8=81-a_1\\ a_1=16\\ a_8=81-a_1\\ a_1+7d=81-a_1\\ 7d+a_1-a_1=81-a_1-a_1\\ 7d=81-2a_1=81-32=49\\ 7d=49\\ d=\frac{49}{7}=7\end{array}$

  $\begin{array}{l}{a}_1=16;d=7\\ a_n=\{16,23,30,37,44,51,58,\mathrm{65.....}\}\end{array}$

  $\text{Sum}\text{of}\text{first}\text{8}\text{terms}\text{is}\text{:16+23+30+37+44+51+58+65=324}$

Thus, $a_n=\{16,23,30,37,44,51,58,\mathrm{65.....}\}$