Question

In Exercises, find a plausible formula for the general term of the given sequence.

$\left\{\frac{1}{3},\frac{2}{9},\frac{1}{9},\frac{4}{81},\frac{5}{243},\dots \right\}$

Short answer

The plausible formula for the general term of the sequence is$a_k=\frac{k}{3^k}.$

Step-by-step solution

Step 1. Given information.

The given sequence is$\left\{\frac{1}{3},\frac{2}{9},\frac{1}{9},\frac{4}{81},\frac{5}{243},\dots \right\}.$

Step 2. plausible formula for the general term. 

write the sequence in a pattern.

$$\begin{gathered} \left\{\frac{1}{3},\frac{2}{9},\frac{1}{9},\frac{4}{81},\frac{5}{243},\dots \right\}=\left\{\frac{1}{3^1},\frac{2}{3^2},\frac{3}{3^3},\frac{4}{3^4},\frac{5}{3^5},\dots \right\} \\ \left\{\frac{1}{3},\frac{2}{9},\frac{1}{9},\frac{4}{81},\frac{5}{243},\dots \right\}={\left\{\frac{k}{3^k}\right\}}_{k=1}^{\infty } \end{gathered}$$

here for every term, the numerator is the natural number k and the denominator is $3^k.$

So plausible formula for the general term of the sequence is$a_k=\frac{k}{3^k}.$