Chapter 12, Problem 27RE

To determine

To find: The sum of the given expression.

Answer to Problem 27RE

The sum of the expression is $1125$ .

Explanation of Solution

Given:

The given expression is,

  ${\displaystyle \sum _{k=1}^{30}\left(3k-9\right)}$

Calculation:

Consider the given expression is,

  ${\displaystyle \sum _{k=1}^{30}\left(3k-9\right)}$

Consider the value of first term is,

  $\begin{array}{c}{a}_1=3\cdot 1-9\\ =-6\end{array}$

Consider the value of second term is,

  $\begin{array}{c}{a}_2=3\cdot 2-9\\ =-3\end{array}$

Consider the value of third term is,

  $\begin{array}{c}{a}_3=3\cdot 3-9\\ =0\end{array}$

Consider the last term of the series is,

  $\begin{array}{c}{a}_{30}=3\cdot 30-9\\ =81\end{array}$

Consider the formula for the sum of arithmetic series is,

  $S_n=\frac{n}{2}\left(a_1+a_n\right)$

Then,

  $\begin{array}{c}{S}_n=\frac{30}{2}\left(-6+81\right)\\ =1125\end{array}$