Chapter 6.6, Problem 23E

To determine

To write next three terms of the geometric sequence:

  $32,8,2,\frac{1}{2},\mathrm{....}$

And also plot the graph of the sequence.

Answer to Problem 23E

The next three terms of given geometric series are: $\frac{1}{8}$ , $\frac{1}{32}$ , and $\frac{1}{128}$ .

  

Explanation of Solution

Given:

The given geometric sequence is:

  $32,8,2,\frac{1}{2},\mathrm{....}$

Concept Used:

• In a geometric sequence, the ration between each pair of consecutive terms is the same, this ratio is called the common ratio.
• Each term of a geometric series is found by multiplying the previous term by the common ratio.

Calculation:

To write next three terms of the geometric sequence:

  $32,8,2,\frac{1}{2},\mathrm{....}$

First find the common ratio of the sequence by dividing second term by first, as

  $\frac{8}{32}=\frac{1}{4}$

Thus, common ration of the given geometric sequence is $\frac{1}{4}$ , that is, each term of the given geometric series can be found by multiplying the previous term by the common ratio $\frac{1}{4}$ .

Thus, next term of $\frac{1}{2}$ will be:

  $\frac{1}{2}\times \frac{1}{4}=\frac{1}{8}$

Similarly, next term of $\frac{1}{8}$ is:

  $\frac{1}{8}\times \frac{1}{4}=\frac{1}{32}$

And, next term of $\frac{1}{32}$ is:

  $\frac{1}{32}\times \frac{1}{4}=\frac{1}{128}$

Thus, next three terms of given geometric series are: $\frac{1}{8}$ , $\frac{1}{32}$ , and $\frac{1}{128}$ .

Now, to plot the graph of given sequence, first make a table of representing terms, say $a_n$ , of the sequence and their respectively place, n, from the above information as:

n	1	2	3	4	5	6	7
$a_n$	32	8	2	$\frac{1}{2}$	$\frac{1}{8}$	$\frac{1}{32}$	$\frac{1}{128}$

Now, using this table plot the graph of the given sequence by plotting the coordinates in above table: $\left(1,32\right),\left(2,8\right),\left(3,2\right),\left(4,\frac{1}{2}\right),\left(5,\frac{1}{8}\right),\left(6,\frac{1}{32}\right),\left(7,\frac{1}{128}\right)$ .

Thus, graph of the sequence is represented by blue and red dots, as