Chapter 12.2, Problem 24E

To determine

To calculate:

The common ratio and next three term of the geometric sequence $t^8,t^5,t^2,\mathrm{...}$

Answer to Problem 24E

The common ratio is $\frac{1}{t^3}$ and next three terms of the geometric sequence are $\frac{1}{t},\frac{1}{t^4},\frac{1}{t^7}$

Explanation of Solution

Given information:

  $t^8,t^5,t^2,\mathrm{...}$

Formula used:

The common ratio is given by the following expression:

  $r=\frac{a_n}{a_{n-1}}$

The $n^{th}$ term of the geometric sequence is given by the following expression:

  $a_n=a_1{r}^{n-1}$

Calculation:

The given geometric sequence is considered as:

  $\begin{array}{l}{a}_1=t^8\\ a_2=t^5\\ a_3=t^2\end{array}$

Now, the common ratio is calculated as:

  $\begin{array}{l}r=\frac{a_2}{a_1}=\frac{t^5}{t^8}=\frac{1}{t^3}\\ r=\frac{a_3}{a_2}=\frac{t^2}{t^5}=\frac{1}{t^3}\end{array}$

The next three terms of the geometric sequence are calculated as:

  $\begin{array}{l}{a}_4=a_1{r}^{4-1}=t^8{\left(\frac{1}{t^3}\right)}^3=\frac{t^8}{t^8}=\frac{1}{t}\\ a_5=a_1{r}^{5-1}=t^8{\left(\frac{1}{t^3}\right)}^4=\frac{t^8}{t^{12}}=\frac{1}{t^4}\\ a_6=a_1{r}^{6-1}=t^8{\left(\frac{1}{t^3}\right)}^5=\frac{t^8}{t^{15}}=\frac{1}{t^7}\end{array}$