Chapter 8.3, Problem 23E

To determine

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

Answer to Problem 23E

nth term of sequence is $a_n=1\cdot 2^{n-1}$ .

  

Explanation of Solution

Given information:

The sequence:

  $\begin{array}{l}{a}_3=4\\ r=2\end{array}$

Formula used:

n^th term of a GP is:

  $a_n=a{r}^{n-1}$

Where,

  $a$ =first term

  $r$ =common ratio

  $\begin{array}{l}{a}_3=4\\ r=2\end{array}$

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, $n=3$ ,

  $\begin{array}{l}{a}_n=a{r}^{n-1}\\ \\ a_3=a\cdot 2^{3-1}\\ \\ 4=a\cdot 2^2\\ \\ 4=a\cdot 4\end{array}$

  $a=1$

Rule for nth term of sequence,

  $\begin{array}{l}a=1\\ r=2\end{array}$

  $\begin{array}{l}{a}_n=a{r}^{n-1}\\ \\ a_n=1\cdot 2^{n-1}\end{array}$

nth term of sequence is $a_n=1\cdot 2^{n-1}$ .

Finding first six terms to plot graph

First term, $a=1$

  $\left(1,1\right)$

Second term, $n=2$

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

  $\left(2,2\right)$

third term,

given, $a=4$

  $\left(3,4\right)$

fourth term, $n=4$

  $\begin{array}{l}{a}_3=1\cdot 2^{4-1}\\ \\ a_3=1\cdot 2^3\\ \\ a_3=1\cdot 8\\ \\ a_3=8\end{array}$

  $\left(4,8\right)$

fifth term, $n=5$

  $\begin{array}{l}{a}_5=1\cdot 2^{5-1}\\ \\ a_5=1\cdot 2^4\\ \\ a_5=1\cdot 16\\ \\ a_5=16\end{array}$

  $\left(5,16\right)$

sixth term, $n=6$

  $\begin{array}{l}{a}_6=1\cdot 2^{6-1}\\ \\ a_6=1\cdot 2^5\\ \\ a_6=1\cdot 32\\ \\ a_6=32\end{array}$

  $\left(6,32\right)$

Plotting the points with above points: