Chapter 11.3, Problem 10WE

To determine

To calculate:

The value of $t_8$ the following geometric sequence:

  $1,-3,9,-27,\mathrm{....}$

Answer to Problem 10WE

The value of $t_8$ the geometric sequence is $-2187$ .

Explanation of Solution

Given information:

  $1,-3,9,-27,\mathrm{....}$

Formula used:

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

  $t_n=t_1{r}^{n-1}$

Where,

First term: $t_1$

Common difference: $r$

Calculation:

Consider the sequence as:

  $\begin{array}{l}{t}_1=1\\ t_2=-2\\ t_3=9\\ t_4=-27\end{array}$

Now, the common ratio is calculated as:

  $\begin{array}{l}r=\frac{t_2}{t_1}=\frac{-3}{1}=-3\\ r=\frac{t_3}{t_2}=\frac{9}{-3}=-3\\ r=\frac{t_4}{t_3}=\frac{-27}{9}=-3\end{array}$

Know that:

First term: $t_1=1$

Common difference: $r=-3$

  $t_8$ term for given geometric sequence is calculated as:

  $\begin{array}{l}{t}_8=t_1{r}^{n-1}\\ t_8=\left(1\right){\left(-3\right)}^{8-1}\\ t_8={\left(-3\right)}^7\\ t_8=-2187\end{array}$