Chapter 9.1, Problem 46E

To determine

The expression for the apparent nth term

Answer to Problem 46E

  $a_n=\frac{{\left(2\right)}^{n-1}}{3^n}$

Explanation of Solution

Given:

The given sequence is

  $\frac{1}{3},\frac{2}{9},\frac{4}{27},\frac{8}{81},\mathrm{...............}$

Calculation:

Consider that $a_i$ is the i^th term of the sequence

For $n=1,2,3,4,5,\mathrm{..................}n$

The terms are $\frac{1}{3},\frac{2}{9},\frac{4}{27},\frac{8}{81},\mathrm{.............................................}{a}_n$

The terms can be written as

  $\begin{array}{l}=\frac{{\left(2\right)}^{1-1}}{3^1},\frac{{\left(2\right)}^{2-1}}{3^2},\frac{{\left(2\right)}^{3-1}}{3^3},\frac{{\left(2\right)}^{4-1}}{3^4},\mathrm{...............}{a}_n\\ =\frac{{\left(2\right)}^0}{3},\frac{{\left(2\right)}^1}{9},\frac{{\left(2\right)}^2}{27},\frac{{\left(2\right)}^3}{81},\mathrm{...............}{a}_n\end{array}$

Therefore, the general term can be written as

  $a_n=\frac{{\left(2\right)}^{n-1}}{3^n}$