Chapter 12.3, Problem 33AYU

To determine

To find: The $n\text{th}$ term of the geometric sequence.

Answer to Problem 33AYU

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

Explanation of Solution

Given information:

The sequence is $-3,1,\frac{-1}{3},\frac{1}{9}\mathrm{.......}$ .

Calculation:

The first term of a sequence is denoted as $a_1$ and the second term of the sequence is denoted by $a_2$ .

Calculate the common ratio.

  $\begin{array}{c}r=\frac{a_2}{a_1}\\ =\frac{1}{-3}\\ =-\frac{1}{3}\end{array}$

The formula for nth term of geometric sequence is given by,

  $a_n=a_1{r}^{n-1}$

Calculate the $n\text{th}$ term.

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

Therefore, the $n\text{th}$ term of the sequence is $a_n=\left(-3\right){\left(-\frac{1}{3}\right)}^{n-1}$ .