Chapter 9.1, Problem 67E

To determine

To find the sum:

  ${\displaystyle \sum _{i=0}^{4}3i^2}$

Answer to Problem 67E

  ${\displaystyle \sum _{i=0}^{4}3i^2}=90$

Explanation of Solution

Given:

  ${\displaystyle \sum _{i=0}^{4}3i^2}$

Concept Used:

  ${\displaystyle \sum _{i=0}^{n}f\left(x_i\right)}=f\left(x_0\right)+f\left(x_1\right)+\mathrm{......}+f\left(x_n\right)$

Calculation:

In order todetermine the sum:

  ${\displaystyle \sum _{i=0}^{4}3i^2}$

Expand this summation and evaluate.

As,

  $\begin{array}{c}{\displaystyle \sum _{i=0}^{4}3i^2}=3{\left(0\right)}^2+3{\left(1\right)}^2+3{\left(2\right)}^2+3{\left(3\right)}^2+3{\left(4\right)}^2\\ =0+3+3\times 4+3\times 9+3\times 16\\ =3+12+27+48\\ =90\end{array}$

Thus,

  ${\displaystyle \sum _{i=0}^{4}3i^2}=90$