Chapter 8.1, Problem 48E

To determine

To write an expression for the apparent $n$ th term of the sequence.

Answer to Problem 48E

The apparent $n$ th term of the sequence is $a_n=\frac{{\left(-2\right)}^{n-1}}{3^n}$

Explanation of Solution

Given information:

Given sequence

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

Calculation:

Arrange the values of $n$ and the terms of the sequence in atable.

$n$	1	2	3	4
Terms	$\frac{1}{3}=\frac{{\left(-2\right)}^{1-1}}{3^1}$	$-\frac{2}{9}=\frac{{\left(-2\right)}^{2-1}}{3^2}$	$\frac{4}{27}=\frac{{\left(-2\right)}^{3-1}}{3^3}$	$-\frac{8}{81}=\frac{{\left(-2\right)}^{4-1}}{3^4}$

If we observe the table, we can see that the apparent $n$ th term of the sequence is $a_n=\frac{{\left(-2\right)}^{n-1}}{3^n}$