Chapter 12, Problem 13RE

a.

To determine

To find the first five terms of the sequence.

Answer to Problem 13RE

  $a_1=\frac{3}{4},a_2=\frac{9}{8},a_3=\frac{27}{16},a_4=\frac{81}{32},a_5=\frac{243}{64}$

Explanation of Solution

Given information:

  $a_n=\frac{3^n}{2^{n+1}}$

Calculation:

Substitute $n=1,2,3,4,5$ in the given sequence and evaluate the expression to find the first five terms as shown below.

  $\begin{array}{c}{a}_1=\frac{3^1}{2^{1+1}}=\frac{3}{2^2}=\frac{3}{4}\\ a_2=\frac{3^2}{2^{2+1}}=\frac{3^2}{2^3}=\frac{9}{8}\\ a_3=\frac{3^3}{2^{3+1}}=\frac{27}{2^4}=\frac{27}{16}\\ a_4=\frac{3^4}{2^{4+1}}=\frac{81}{2^5}=\frac{81}{32}\\ a_5=\frac{3^5}{2^{5+1}}=\frac{243}{2^6}=\frac{243}{64}\end{array}$

b.

To determine

To graph the terms found in part (a).

Explanation of Solution

Given information:

  $a_n=\frac{3^n}{2^{n+1}}$

Graph:

The terms found in part (a) are $a_1=\frac{3}{4},a_2=\frac{9}{8},a_3=\frac{27}{16},a_4=\frac{81}{32},a_5=\frac{243}{64}$ . These can be graphed as shown below.

  

c.

To determine

To find the fifth partial sum of the sequence.

Answer to Problem 13RE

  $\frac{155}{32}$

Explanation of Solution

Given information:

  $a_n=\frac{3^n}{2^{n+1}}$

Calculation:

The first five terms of the sequence are:

  $a_1=\frac{3}{4},a_2=\frac{9}{8},a_3=\frac{27}{16},a_4=\frac{81}{32},a_5=\frac{243}{64}$

So, the fifth partial sum is;

  $\begin{array}{c}{S}_5=a_1+a_2+a_3+a_4+a_5\\ =\frac{3}{4}+\frac{9}{8}+\frac{27}{16}+\frac{81}{32}+\frac{243}{64}\\ =\frac{633}{64}\end{array}$

d.

To determine

To find the fifth partial sum of the sequence.

Answer to Problem 13RE

Geometric, common ratio is $\frac{3}{2}$ .

Explanation of Solution

Given information:

  $a_n=\frac{3^n}{2^{n+1}}$

Calculation:

The first five terms of the sequence are:

  $a_1=\frac{3}{4},a_2=\frac{9}{8},a_3=\frac{27}{16},a_4=\frac{81}{32},a_5=\frac{243}{64}$

Observe that $\frac{a_2}{a_1}=\frac{a_3}{a_2}=\frac{a_4}{a_3}=\frac{a_5}{a_4}=\frac{3}{2}$ .

So, the given sequence is a geometric sequence with a common ratio of $\frac{3}{2}$ .