Chapter 9.3, Problem 19E

To determine

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

Answer to Problem 19E

The first five terms of the geometric sequence are $3,3\sqrt{5,}15,15\sqrt{5},75$ ..

Explanation of Solution

Given information:

The given sequence,

  $a_1=3,r=\sqrt{5}$

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

  $\begin{array}{l}{a}_1=3\\ r=\sqrt{5}\end{array}$

To find the second term, multiply first term $\sqrt{5}$ as shown below.

  $\begin{array}{c}{a}_2=3\left(\sqrt{5}\right)\\ =3\sqrt{5}\end{array}$

The 3^rd term can be calculated as

  $\begin{array}{c}{a}_3=3{\left(\sqrt{5}\right)}^2\\ =15\end{array}$

The 4^th term can be calculated as

  $\begin{array}{c}{a}_4=3{\left(\sqrt{5}\right)}^3\\ =15\sqrt{5}\end{array}$

The 5^th term can be calculated as

  $\begin{array}{c}{a}_5=3{\left(\sqrt{5}\right)}^4\\ =75\end{array}$

Hence, the first five terms of the geometric sequence are

  $3,3\sqrt{5,}15,15\sqrt{5},75$