Chapter 8.3, Problem 24E

To determine

To find: rule for nth term of sequence and plot the graph for first six terms.

Answer to Problem 24E

nth term of sequence is $a_n=3\cdot 3^{n-1}$ .

  

Explanation of Solution

Given information:

The sequence:

  $\begin{array}{l}{a}_3=27\\ r=3\end{array}$

Formula used:

n^th term of a GP is:

  $a_n=a{r}^{n-1}$

Where,

  $a$ =first term

  $r$ =common ratio

  $\begin{array}{l}{a}_3=27\\ r=3\end{array}$

Recall that n^th term of a GP is:

  $a_n=a{r}^{n-1}$

Where,

  $a$ =first term

  $r$ =common ratio

  $n$ =position of term in sequence

As the third term of sequence is given, $n=3$ ,

  $\begin{array}{l}{a}_n=a{r}^{n-1}\\ \\ a_3=a\cdot 3^{3-1}\\ \\ 27=a\cdot 3^2\\ \\ 27=a\cdot 9\end{array}$

  $a=3$

Rule for nth term of sequence,

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

  $\begin{array}{l}{a}_n=a{r}^{n-1}\\ \\ a_n=3\cdot 3^{n-1}\end{array}$

nth term of sequence is $a_n=3\cdot 3^{n-1}$ .

Finding first six terms to plot graph

First term, $a=3$

  $\left(1,3\right)$

Second term, $n=2$

  $\begin{array}{l}{a}_2=3\cdot 3^{2-1}\\ \\ a_2=3\cdot 3^1\\ \\ a_2=9\end{array}$

  $\left(2,9\right)$

third term,

given, $a=4$

  $\left(3,27\right)$

fourth term, $n=4$

  $\begin{array}{l}{a}_n=3\cdot 3^{n-1}\\ \\ a_4=3\cdot 3^{4-1}\\ \\ a_4=3\cdot 3^3\\ \\ a_4=3\cdot 27\end{array}$

  $a_4=81$

  $\left(4,81\right)$

fifth term, $n=5$

  $\begin{array}{l}{a}_5=3\cdot 3^{5-1}\\ \\ a_5=3\cdot 3^4\\ \\ a_5=3\cdot 81\\ \\ a_5=243\end{array}$

  $\left(5,243\right)$

sixth term, $n=6$

  $\begin{array}{l}{a}_6=3\cdot 3^{6-1}\\ \\ a_6=3\cdot 3^5\\ \\ a_6=3\cdot 243\\ \\ a_6=729\end{array}$

  $\left(6,729\right)$

Plotting the points with above points: