Chapter 8.1, Problem 79E

To determine

To match: the given sequence with its graph.

Answer to Problem 79E

The given sequence matches with graph (d).

Explanation of Solution

Given:

A sequence is given as $a_n=\frac{2^{n+2}}{2n!}$ .

Given graphs are

  

Formula used:

Factorial of a number n can be simplified as

  $n!=n\left(n-1\right)\left(n-2\right)\left(n-3\right)\mathrm{...3}\cdot 2\cdot 1$

Where $n$ is a positive integer.

Calculation:

Consider the n th term of the sequence.

  $a_n=\frac{2^{n+2}}{2n!}$

Now, substitute 1 for n in $a_n=\frac{2^{n+2}}{2n!}$ for the first term of the sequence.

  $\begin{array}{c}{a}_1=\frac{2^{1+2}}{2\cdot 1!}\\ =\frac{2^3}{2\cdot 1}\\ =4\end{array}$

Now, construct the following table for first ten terms of the sequence.

n	1	2	3	4	5	6	7	8	9	10
Terms	4	4	2.67	1.33	0.533	0.178	0.05	0.0127	0.0028	0.005

Now, plot the terms and the graph looks like as shown:

  

So, the graph matches with graph (d).