Question

Evaluating a Sum Find the sum.

$\sum _{j=1}^{100}{\left(-1\right)}^j$

Short answer

The sum is 0.

Step-by-step solution

Step 1.  Given

The given sequence is$\sum _{j=1}^{100}{\left(-1\right)}^j.$

Step 2.  To determine

We have to find thesum of the given sequence.

Step 3.  Calculation

We’ll plug the values of j and get rid of the summation sign to simplify:

$\begin{array}{l}\sum _{j=1}^{100}{\left(-1\right)}^j\\ ={\left(-1\right)}^1+{\left(-1\right)}^2+{\left(-1\right)}^3+{\left(-1\right)}^4+\mathrm{....}+{\left(-1\right)}^{99}+{\left(-1\right)}^{100}\\ =-1+1-1+1-\mathrm{....}-1+1\\ =\left(-1+1\right)+\left(-1+1\right)+\mathrm{.....}\left(-1+1\right)\\ =0+0+\mathrm{....}+0\\ =0\end{array}$

Hence, the sum is 0.