Question

Find a general term for the sequence whose first five terms are shown 1, −4, 9, −16, 25,...

Short answer

The general term for the given sequence is$a_n={\left(-1\right)}^{n+1}{n}^2$

Step-by-step solution

Step 1. Given information

The first five terms are shown 1, −4, 9, −16, 25,

Step 2. Observing the pattern

$\begin{array}{llllllll}n& 1& 2& 3& 4& 5& \mathrm{...}& n\\ \text{Terms}& 1& -4& 9& -16& 25& \mathrm{...}& \\ \text{Pattern}& {\left(-1\right)}^{1+1}{1}^2& {\left(-1\right)}^{1+2}{2}^2& {\left(-1\right)}^3{3}^3& {\left(-1\right)}^{4+1}{4}^2& {\left(-1\right)}^{5+1}{5}^2& \mathrm{...}& {\left(-1\right)}^{n+1}{n}^2\end{array}$

Step 3. Conclusion

The numbers are powers of 2. The signs are alternating, with even $n$negative.

Therefore, the general term for the given sequence islocalid="1645103580853" $a_n={\left(-1\right)}^{n+1}{n}^2$