Chapter 8.3, Problem 26E

To determine

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

Answer to Problem 26E

nth term of sequence is $a_n=256\cdot {\left(\frac{1}{4}\right)}^{n-1}$ .

  

Explanation of Solution

Given information:

The sequence:

  $\begin{array}{l}{a}_2=64\\ r=\frac{1}{4}\end{array}$

Formula used:

n^th term of a GP is:

  $a_n=a{r}^{n-1}$

Where,

  $a$ =first term

  $r$ =common ratio

  $\begin{array}{l}{a}_2=64\\ r=\frac{1}{4}\end{array}$

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 second term of sequence is given, $n=2$ ,

  $\begin{array}{l}{a}_n=a{r}^{n-1}\\ \\ a_2=a\cdot {\left(\frac{1}{4}\right)}^{2-1}\\ \\ 64=a\cdot \left(\frac{1}{4}\right)\\ \\ a=256\end{array}$

Rule for nth term of sequence,

  $\begin{array}{l}a=60\\ r=\frac{1}{4}\end{array}$

  $\begin{array}{l}{a}_n=a{r}^{n-1}\\ \\ a_n=60\cdot {\left(\frac{1}{2}\right)}^{n-1}\end{array}$

nth term of sequence is $a_n=256\cdot {\left(\frac{1}{4}\right)}^{n-1}$ .

Finding first six terms to plot graph

First term, $a=256$

  $\left(1,256\right)$

Second term, $n=2$

Given, $a_2=64$

  $\left(2,64\right)$

Third term, $n=3$

  $\begin{array}{l}{a}_3=256\cdot {\left(\frac{1}{4}\right)}^{3-1}\\ \\ a_3=256\cdot {\left(\frac{1}{4}\right)}^2\\ \\ a_3=256\cdot \frac{1}{16}\\ \\ a_3=16\end{array}$

  $\left(4,16\right)$

fourth term, $n=4$

  $\begin{array}{l}{a}_4=256\cdot {\left(\frac{1}{4}\right)}^{4-1}\\ \\ a_4=256\cdot {\left(\frac{1}{4}\right)}^3\\ \\ a_4=256\cdot \frac{1}{64}\\ \\ a_4=4\end{array}$

  $a_4=\frac{15}{2}$

  $\left(4,4\right)$

fifth term, $n=5$

  $\begin{array}{l}{a}_5=256\cdot {\left(\frac{1}{4}\right)}^{5-1}\\ \\ a_5=256\cdot {\left(\frac{1}{4}\right)}^4\\ \\ a_5=256\cdot \frac{1}{256}\\ \\ a_5=1\end{array}$

  $\left(5,1\right)$

sixth term, $n=6$

  $\begin{array}{l}{a}_6=256\cdot {\left(\frac{1}{4}\right)}^{6-1}\\ \\ a_6=256\cdot {\left(\frac{1}{4}\right)}^5\\ \\ a_6=256\cdot \frac{1}{1024}\\ \\ a_6=\frac{1}{4}\end{array}$

  $\left(6,\frac{1}{4}\right)$

Plotting the points with above points: