Question

Find a general term for the sequence whose first five terms are shown$\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{16},\frac{1}{32},\dots$

Short answer

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

Step-by-step solution

Step 1. Given information

$\begin{array}{l}\text{Thegivensequenceis}\\ \frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{16},\frac{1}{32},\dots \\ \text{Wehavetofindthegeneraltermforthegivensequence.}\end{array}$

Step 2. Observing the pattern

$\begin{array}{llllllll}n& 1& 2& 3& 4& 5& \mathrm{...}& n\\ \text{Terms}& \frac{1}{2}& \frac{1}{4}& \frac{1}{8}& \frac{1}{16}& \frac{1}{32}& \mathrm{...}& \\ \text{Pattern}& \frac{1}{2^1}& \frac{1}{2^2}& \frac{1}{2^3}& \frac{1}{2^4}& \frac{1}{2^5}& \mathrm{...}& \frac{1}{2^n}\end{array}$

Step 3. Conclusion

We can observe that the numerators are all 1 and the denominators are powers of 2.

Therefore, the general term will be$a_n=\frac{1}{2^n}$