Chapter 8.3, Problem 40E

To determine

To find: an expression for nth term and the 7^h term of the geometric sequence.

Answer to Problem 40E

An expression for nth term is $a_n=3{\left(12\right)}^{n-1}$ .

The 7^thterm of the geometric sequence will be $a_7=128994509$ .

Explanation of Solution

Given information:

Three terms of the sequence are given by,

  $3,36,432,\mathrm{...}$

Concept used:

The general term of geometric sequence is given by formula

  $a_n=a_1{r}^{n-1}$

Here, $a_1$ is the first term and $r$ is common ratio.

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:

Consider the given sequence.

  $3,36,432,\mathrm{...}$

Now $a_2=a_1{r}^{2-1}$

  $\begin{array}{l}36=3r\\ r=\frac{36}{3}\\ r=\mathrm{12...............}(1)\end{array}$

Now, obtain the expression for nth term as shown:

  $\begin{array}{c}{a}_n=a_1{r}^{n-1}\\ =3\times {\left(12\right)}^{n-1}\end{array}$

Thus, an expression for nth term is $a_n=3{\left(12\right)}^{n-1}$ .

Now, calculate the value of $a_7$ by using the formula as shown:

  $\begin{array}{c}{a}_7=a_1{r}^{7-1}\\ a_7=a_1{r}^6\\ a_7=3\times {\left(12\right)}^6\\ a_7=128994509\end{array}$

Hence, the 7^thterm will be $a_7=128994509$ .