Chapter 8.3, Problem 17E

To determine

To write: the first five terms of the geometric sequence.

Answer to Problem 17E

The first five terms of the geometric sequence are $6,18,54,162,486$ .

Explanation of Solution

Given information:

The given sequence,

  $a_1=6,r=3$

Concept used:

The $n$ th term of geometric sequence has the form $a_1=a_1{r}^{n-1}$ where $r$ is the common ratio of the consecutive terms of the sequence.

Every geometric sequence can be written in the following form.

  $a_1=a_1,a_2=a_{1}r,a_3=a_1{r}^2,a_4=a_1{r}^3,\mathrm{...},a_n=a_1{r}^{n-1},\mathrm{..}$

Calculation:

The common ratio and the first term of the sequence is given by

  $\begin{array}{l}{a}_1=6\\ r=3\end{array}$

Substituting the values of $r$ and $a_1$ . So, the formula is of the form,

  $\begin{array}{c}{a}_1=6\\ a_2=a_1{r}^{2-1}\\ =6\left(3\right)\\ =18\end{array}$

The 3^rd term can be calculated as

  $\begin{array}{c}{a}_3=a_1{r}^{3-1}\\ =6{\left(3\right)}^2\\ =54\end{array}$

The 4^th term can be calculated as

  $\begin{array}{c}{a}_4=a_1{r}^{4-1}\\ =6{\left(3\right)}^3\\ =162\end{array}$

The 5^th term can be calculated as

  $\begin{array}{c}{a}_5=a_1{r}^{5-1}\\ =6{\left(3\right)}^4\\ =486\end{array}$

Hence, the first five terms of the geometric sequence are

  $6,18,54,162,486$