Chapter 6.5, Problem 21E

To determine

To write next three terms of the geometric sequence:

  $81,-27,9,-3,\mathrm{....}$

And also plot the graph of the sequence.

Answer to Problem 21E

The next three terms of given geometric series are: 1, $-\frac{1}{3}$ , and $\frac{1}{9}$

Explanation of Solution

Given:

The given geometric sequence is:

  $81,-27,9,-3,\mathrm{....}$

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:

  $81,-27,9,-3,\mathrm{....}$

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

  $\frac{-27}{81}=-\frac{1}{3}$

Thus, common ration of the given geometric sequence is $-\frac{1}{3}$ , that is, each term of the given geometric series can be found by multiplying the previous term by the common ratio $-\frac{1}{3}$ .

Thus, next term of $-3$ will be:

  $-3\times \left(-\frac{1}{3}\right)=1$

Similarly, next term of 1 is:

  $1\times \left(-\frac{1}{3}\right)=-\frac{1}{3}$

And, next term of $-\frac{1}{3}$ is:

  $-\frac{1}{3}\times \left(-\frac{1}{3}\right)=\frac{1}{9}$

Thus, next three terms of given geometric series are: 1, $-\frac{1}{3}$ , and $\frac{1}{9}$ .

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
$a_n$	81	$-27$	9	$-3$	1	$-\frac{1}{3}$	$\frac{1}{9}$

Now, using this table plot the graph of the given sequence by plotting the coordinates in above table: $\left(1,81\right),\left(2,-27\right),\left(3,9\right),\left(4,-3\right),\left(5,1\right),\left(6,-\frac{1}{3}\right),\left(7,\frac{1}{9}\right)$ .

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