Question

Find a general term for the sequence whose first five terms are shown. −3, 9, −27, 81, −243, ...

Short answer

The general term for the given sequence is$a_n={\left(-1\right)}^n{3}^n$

Step-by-step solution

Step 1. Given information

The first five terms are shown. −3, 9, −27, 81, −243,...

Step 2. Observing the pattern

$\begin{array}{llllllll}n& 1& 2& 3& 4& 5& \mathrm{...}& n\\ \text{Terms}& -3& 9& -27& 81& -243& \mathrm{...}& \\ \text{Pattern}& {\left(-1\right)}^1{3}^1& {\left(-1\right)}^2{3}^2& {\left(-1\right)}^3{3}^3& {\left(-1\right)}^4{3}^4& {\left(-1\right)}^5{3}^5& \mathrm{...}& {\left(-1\right)}^n{3}^n\end{array}$

Step 3. Conclusion

The numbers are powers of 3. The signs are alternating, with odd $n$negative.

Therefore,

The general term islocalid="1645103689408" $a_n={\left(-1\right)}^n{3}^n$