Question

Find a general term for the sequence whose first five terms are shown.

$\frac{1}{1},\frac{1}{4},\frac{1}{9},\frac{1}{16},\frac{1}{25},\dots$

Short answer

The general term for the given sequence is$a_n=\frac{1}{n^2}$.

Step-by-step solution

Step 1. Given information

$\begin{array}{l}\text{Thegivensequenceis}\\ \frac{1}{1},\frac{1}{4},\frac{1}{9},\frac{1}{16},\frac{1}{25},\dots \\ \text{Wehavetofindthegeneraltermforthegivensequence.}\end{array}$

Step 2. Observing the pattern 

$\begin{array}{llllllll}n& 1& 2& 3& 4& 5& \mathrm{...}& n\\ \text{Terms}& \frac{1}{1}& \frac{1}{4}& \frac{1}{9}& \frac{1}{16}& \frac{1}{25}& \mathrm{...}& \\ \text{Pattern}& \frac{1}{1^2}& \frac{1}{2^2}& \frac{1}{3^2}& \frac{1}{4^2}& \frac{1}{5^2}& \mathrm{...}& \frac{1}{n^2}\end{array}$

Step 3. Conclusion

From the table in the above step, we can observe that the numerators are all 1. The denominator is the square of the number of terms.

Therefore,

The general term of the given sequence can be written as$a_n=\frac{1}{n^2}.$