Question

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

−6, 11, −16, 21, −26, …

Short answer

The general term for the sequence whose first five terms are −6, 11, −16, 21, −26, … is$a_n={\left(-1\right)}^n\left(5n+1\right).$

Step-by-step solution

Step 1. Given information

$\begin{array}{l}\text{Thefirstfivetermsofthesequenceare}\\ -6,11,-16,21,-26,\dots \\ \text{Wehavetofindthegeneraltermforthesequence.}\end{array}$

Step 2. Observing the pattern

Now,

$\begin{array}{cccccccc}n:& 1& 2& 3& 4& 5& \mathrm{...}& n\\ \text{Terms:}& -6& 11& -16& 21& -26& \mathrm{...}& \\ \text{Pattern}& {\left(-1\right)}^1\left(5\cdot 1+1\right)& {\left(-1\right)}^2\left(5\cdot 2+1\right)& {\left(-1\right)}^3\left(5\cdot 3+1\right)& {\left(-1\right)}^4\left(5\cdot 4+1\right)& {\left(-1\right)}^5\left(5\cdot 5+1\right)& \mathrm{...}& {\left(-1\right)}^n\left(5n+1\right)\end{array}$

Step 3. Conclusion

From the above table, we can observe that the general terms will be$a_n={\left(-1\right)}^n\left(5n+1\right).$