Question

In Problems 51–60, write out each sum.

$\sum _{k=1}^n\frac{k^2}{2}$

Short answer

The required sum is$\sum _{k=1}^n\frac{k^2}{2}=\left\{\frac{1}{2}+2+\frac{9}{2}+......+\frac{n^2}{2}\right\}$.

Step-by-step solution

Step 1. Write the given information.

The given summation notation is:

$\sum _{k=1}^n\frac{k^2}{2}$

Step 2. Write out the sum.

To get the sum, we need to substitute $1,2,3,......,n$for k,in the given sequence $\sum _{k=1}^n\frac{k^2}{2}$,

$$\begin{gathered} \sum _{k=1}^n\frac{k^2}{2}=\frac{1^2}{2}+\frac{2^2}{2}+\frac{3^2}{2}+.......+\frac{n^2}{2} \\ =\frac{1}{2}+2+\frac{9}{2}+......+\frac{n^2}{2} \end{gathered}$$

Therefore,$\sum _{k=1}^n\frac{k^2}{2}=\left\{\frac{1}{2}+2+\frac{9}{2}+......+\frac{n^2}{2}\right\}$.