Chapter 8.3, Problem 116E

To determine

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

Answer to Problem 116E

The first five terms of the geometric sequence are $\frac{1}{4},\frac{7x}{4},\frac{49x^2}{4},\frac{343x^3}{4},\frac{2401x^4}{4}$ respectively.

Explanation of Solution

Given information:

The given sequence,

  $a_1=\frac{1}{4},r=7x$

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.

  $\begin{array}{l}{a}_1=3\\ r=\frac{x}{2}\end{array}$

To find the second term, multiply first term $\frac{x}{2}$ as shown below.

  $\begin{array}{c}{a}_2=a_1{r}^{2-1}\\ =\frac{1}{4}\left(7x\right)\\ =\frac{7x}{4}\end{array}$

The 3^rd term can be calculated as

  $\begin{array}{c}{a}_3=a_1{r}^{3-1}\\ =\frac{1}{4}{\left(7x\right)}^2\\ =\frac{49x^2}{4}\end{array}$

The 4^th term can be calculated as

  $\begin{array}{c}{a}_4=a_1{r}^{4-1}\\ =\frac{1}{4}{\left(7x\right)}^3\\ =\frac{343x^3}{4}\end{array}$

The 5^th term can be calculated as

  $\begin{array}{c}{a}_5=a_1{r}^{5-1}\\ =\frac{1}{4}{\left(7x\right)}^4\\ =\frac{2401x^4}{4}\end{array}$

Hence, the first five terms of the geometric sequence are

  $\frac{1}{4},\frac{7x}{4},\frac{49x^2}{4},\frac{343x^3}{4},\frac{2401x^4}{4}$