Chapter 6.6, Problem 20E

To determine

To write next three terms of the geometric sequence:

  $2500,500,100,20,\mathrm{....}$

And also plot the graph of the sequence.

Answer to Problem 20E

The next three terms of given geometric series are: 4, 0.8, and 0.16.

  

Explanation of Solution

Given:

The given geometric sequence is:

  $2500,500,100,20,\mathrm{....}$

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:

  $2500,500,100,20,\mathrm{....}$

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

  $\begin{array}{c}\frac{500}{2500}=\frac{1}{5}\\ =0.2\end{array}$

Thus, common ration of the given geometric sequence is $0.2$ , that is, each term of the given geometric series can be found by multiplying the previous term by the common ratio $0.2$.

Thus, next term of 20 will be:

  $\begin{array}{c}20\times 0.2=4.0\\ =4\end{array}$

Similarly, next term of 4 is:

  $4\times 0.2=0.8$

And, next term of $0.8$ is:

  $0.8\times 0.2=0.16$

Thus, next three terms of given geometric series are: 4, 0.8, and 0.16.

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
$a_n$	2500	500	100	20	4	$0.8$	$0.16$

Now, using this table plot the graph of the given sequence by plotting the coordinates in above table: $\left(1,2500\right),\left(2,500\right),\left(3,100\right),\left(4,20\right),\left(5,4\right),\left(6,0.8\right),\left(7,0.16\right)$.

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