Chapter 9.9, Problem 44SR

To determine

The equation for nth term and 7^th term of geometric sequence

Answer to Problem 44SR

The equation for nth term of the sequence is $a_n=99{\left(-\frac{1}{3}\right)}^{n-1}$

7^th term is $\frac{11}{81}$

Explanation of Solution

Given information:

Sequence $99,-33,11,\mathrm{...}$

Concept Used:

The equation for nth term is $a_n=a_1{\left(r\right)}^{n-1}$

The ratio is given by the formula

  $r=\frac{a_2}{a_1}=\frac{a_3}{a_2}$

Calculation:

In order to find the equation for nth term and 7^th terms in the sequence simplify further as shown below,

  $\begin{array}{c}{a}_n=a_1{\left(r\right)}^{n-1}\\ a_1=99\\ r=\frac{a_2}{a_1}=\frac{-33}{99}=-\frac{1}{3}\\ \text{equation for nth term is}\\ a_n=99{\left(-\frac{1}{3}\right)}^{n-1}\\ a_7=99{\left(-\frac{1}{3}\right)}^{7-1}\\ =99{\left(-\frac{1}{3}\right)}^6\\ =99\times \frac{1}{729}\\ a_7=\frac{11}{81}\end{array}$

Thus, the equation for nth term of the sequence is $a_n=99{\left(-\frac{1}{3}\right)}^{n-1}$

7^th term is $\frac{11}{81}$