Chapter 11.3, Problem 11WE

To determine

To calculate:

The value of $t_{11}$ the following geometric sequence:

  $40,-20,10,-5,\mathrm{....}$

Answer to Problem 11WE

The value of $t_{11}$ the geometric sequence is $40960$ .

Explanation of Solution

Given information:

  $40,-20,10,-5,\mathrm{....}$

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:

  $\begin{array}{l}{t}_1=40\\ t_2=-20\\ t_3=10\\ t_4=-5\end{array}$

Now, the common ratio is calculated as:

  $\begin{array}{l}r=\frac{t_2}{t_1}=\frac{-20}{40}=-2\\ r=\frac{t_3}{t_2}=\frac{-20}{10}=-2\\ r=\frac{t_4}{t_3}=\frac{10}{-5}=-2\end{array}$

Know that:

First term: $t_1=40$

Common difference: $r=-2$

  $t_{11}$ term for given geometric sequence is calculated as:

  $\begin{array}{l}{t}_{11}=t_1{r}^{n-1}\\ t_{11}=\left(40\right){\left(-2\right)}^{11-1}\\ t_{11}=40{\left(-2\right)}^{10}\\ t_{11}=40960\end{array}$