Chapter 9, Problem 7CT

To determine

To calculate: The value of the expression

  ${\displaystyle \sum _{n=1}^9\left(12n-7\right)}$ .

Answer to Problem 7CT

The value of the expression ${\displaystyle \sum _{n=1}^9\left(12n-7\right)}$ is $477$ .

Explanation of Solution

Given information:

The expression ${\displaystyle \sum _{n=1}^9\left(12n-7\right)}$ .

Formula used:

Summation property, ${\displaystyle \sum \left(a-b\right)={\displaystyle \sum a}}-{\displaystyle \sum b}$ and ${\displaystyle \sum _{n=1}^{m}an}=a{\displaystyle \sum _{n=1}^{m}n}$ .

The sum of first n terms of sequence is given by $\frac{n\left(n+1\right)}{2}$ .

Calculation:

Consider expression ${\displaystyle \sum _{n=1}^9\left(12n-7\right)}$ . Apply the summation property, ${\displaystyle \sum \left(a-b\right)={\displaystyle \sum a}}-{\displaystyle \sum b}$ and ${\displaystyle \sum _{n=1}^{m}an}=a{\displaystyle \sum _{n=1}^{m}n}$ to obtain,

  $\begin{array}{c}{\displaystyle \sum _{n=1}^9\left(12n-7\right)}={\displaystyle \sum _{n=1}^{9}12}n-{\displaystyle \sum _{n=1}^{9}7}\\ =12{\displaystyle \sum _{n=1}^{9}n}-{\displaystyle \sum _{n=1}^{9}7}\end{array}$

Expand the expression as,

  $\begin{array}{c}{\displaystyle \sum _{n=1}^9\left(12n-7\right)}={\displaystyle \sum _{n=1}^{9}12}n-{\displaystyle \sum _{n=1}^{9}7}\\ =12{\displaystyle \sum _{n=1}^{9}n}-{\displaystyle \sum _{n=1}^{9}7}\\ =12\left(1+2+3+\cdots 9\right)-\left(7+7+7+7+7+7+7+7+7\right)\end{array}$

Recall that the sum of first n terms of sequence is given by $\frac{n\left(n+1\right)}{2}$ .

So, sum of first 9 terms is $\frac{9\left(9+1\right)}{2}$ . Apply it,

  $\begin{array}{c}{\displaystyle \sum _{n=1}^9\left(12n-7\right)}=12\left(\frac{9\left(9+1\right)}{2}\right)-63\\ =54\left(10\right)-63\\ =540-63\\ =477\end{array}$

Thus, the value of the expression ${\displaystyle \sum _{n=1}^9\left(12n-7\right)}$ is $477$ .