Chapter 11.3, Problem 7WE

To determine

To calculate:

The value of $t_{10}$ the following geometric sequence:

  $2,6,18,54,\mathrm{....}$

Answer to Problem 7WE

The value of $t_{10}$ the geometric sequence is $39366$ .

Explanation of Solution

Given information:

  $2,6,18,54,\mathrm{....}$

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:

  $\begin{array}{l}{t}_1=2\\ t_2=6\\ t_3=18\\ t_4=54\end{array}$

Now, the common ratio is calculated as:

  $\begin{array}{l}r=\frac{t_2}{t_1}=\frac{6}{2}=3\\ r=\frac{t_3}{t_2}=\frac{18}{6}=3\\ r=\frac{t_4}{t_3}=\frac{54}{18}=3\end{array}$

Know that:

First term: $t_1=2$

Common difference: $r=3$

  $t_{10}$ term for given geometric sequence is calculated as:

  $\begin{array}{l}{t}_{10}=t_1{r}^{n-1}\\ t_{10}=\left(2\right){\left(3\right)}^{10-1}\\ t_{10}=2{\left(3\right)}^9\\ t_{10}=39366\end{array}$