Question

Write each sum using summation notation: −2 + 4 − 6 + 8 − 10.

Short answer

The sum written in summation notation is

$-2+4-6+8-10=\sum _{n=1}^5{\left(-1\right)}^{n}2\cdot n.$

Step-by-step solution

Step 1. Given information

$\begin{array}{l}\text{Thegivensumis}\\ -2+4-6+8-10.\\ \text{Wehavetowritethesumusingsummationnotation.}\end{array}$

Step 2. Find the pattern

Now,

We look for a pattern in the terms.

$\begin{array}{cccccccc}n:& 1& 2& 3& 4& 5& \mathrm{...}& n\\ \text{Terms}& -2& 4& -6& 8& -10& \mathrm{...}& \\ \text{Pattern}& {\left(-1\right)}^{1}2\cdot 1& {\left(-1\right)}^{2}2\cdot 2& {\left(-1\right)}^{3}2\cdot 3& {\left(-1\right)}^{4}2\cdot 4& {\left(-1\right)}^{5}2\cdot 5& \mathrm{...}& {\left(-1\right)}^{n}2\cdot n\end{array}$

Step 3. Conclusion

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

Therefore,

The sum of −2 + 4 − 6 + 8 − 10 can be written as$\sum _{n=1}^5{\left(-1\right)}^{n}2\cdot n$