Chapter 9.8, Problem 11CYU

To determine

The equation and 10^th term of geometric sequence

Answer to Problem 11CYU

The equation for nth term of the sequence is $72{\left(\frac{2}{3}\right)}^{n-1}$

10^th term is $\frac{36864}{19683}$

Explanation of Solution

Given information:

Sequence $72,48,32,\mathrm{...}$

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 next three terms in the sequence simplify further as shown below,

  $\begin{array}{c}{a}_n=a_1{\left(r\right)}^{n-1}\\ a_1=72\\ r=\frac{a_2}{a_1}=\frac{48}{72}=\frac{2}{3}\\ \text{equation for nth term is}\\ a_n=72{\left(\frac{2}{3}\right)}^{n-1}\\ a_{10}=72{\left(\frac{2}{3}\right)}^{10-1}\\ =72\times {\left(\frac{2}{3}\right)}^9\\ a_{10}=72\times \frac{512}{19683}\\ a_{10}=\frac{36864}{19683}\end{array}$

Thus, the equation for nth term of the sequence is $a_n=-1{\left(-5\right)}^{n-1}$

10^th term is $\frac{36864}{19683}$