Question

In the following exercise, find a general term for the sequence whose first five terms are shown.

−1, 8, −27, 64, −125, …

Short answer

The general term for the sequence whose first five terms are −1, 8, −27, 64, −125, … is$a_n={\left(-1\right)}^n{n}^3.$

Step-by-step solution

Step 1. given information

$\begin{array}{l}\text{Thefirstfivetermsofthesequenceare}\\ -1,8,-27,64,-125,\dots \\ \text{Wehavetofindthegeneraltermforthesequence.}\end{array}$

Step 2. Observing the pattern

Now,

$\begin{array}{cccccccc}n:& 1& 2& 3& 4& 5& \mathrm{...}& n\\ \text{Terms:}& -1& 8& -27& 64& -125& \mathrm{...}& \\ \text{Pattern}& {\left(-1\right)}^1{1}^3& {\left(-1\right)}^2{2}^3& {\left(-1\right)}^3{3}^3& {\left(-1\right)}^4{4}^3& {\left(-1\right)}^5{5}^3& \mathrm{...}& {\left(-1\right)}^n{n}^3\end{array}$

Step 3. Conclusion 

From the above table, we can observe that the general terms will be$a_n={\left(-1\right)}^n{n}^3.$