Question

Write each sum using summation notation: 1 − 4 + 9 − 16 + 25.

Short answer

The sum written in summation notation is

$1-4+9-16+25=\sum _{n=1}^5{\left(-1\right)}^{n+1}{n}^2.$

Step-by-step solution

Step 1. Given information

$\begin{array}{l}\text{Thegivensumis}\\ 1-4+9-16+25\\ \text{Wehavetowritethesumusingsummationnotation.}\end{array}$

Step 2. Find the pattern 

Now,

We look for a pattern in the terms.

$\begin{array}{cccccccc}n:& 1& 2& 3& 4& 5& \mathrm{...}& n\\ \text{Terms}& 1& -4& 9& -16& 25& \mathrm{...}& \\ \text{Pattern}& {\left(-1\right)}^{1+1}{1}^2& {\left(-1\right)}^{2+1}{2}^2& {\left(-1\right)}^{3+1}{3}^2& {\left(-1\right)}^{4+1}{4}^2& {\left(-1\right)}^{5+1}{5}^2& \mathrm{...}& {\left(-1\right)}^{n+1}{n}^2\end{array}$

Step 3. Conclusion

From the above table, we can observe that the general term is ${\left(-1\right)}^{n+1}{n}^2$

Therefore,

The sum of $1-4+9-16+25$can be written in the form of summation notation as$\sum _{n=1}^5{\left(-1\right)}^{n+1}{n}^2.$