Chapter 6.5, Problem 54E

a.

To determine

To state: whether the terms $a_1-b_1,a_2-b_2,a_3-b_3$ forms a geometric sequence or not.

Answer to Problem 54E

The terms $a_1-b_1,a_2-b_2,a_3-b_3$ do not form a geometric sequence.

Explanation of Solution

  $\begin{array}{l}{a}_1-b_1=2-1=1\\ a_2-b_2=6-5=1\\ a_3-b_3=18-25=-7\end{array}$

And $\frac{a_2-b_2}{a_1-b_1}=1\text{ and }\frac{a_3-b_3}{a_2-b_2}=-7$

Here common ratio is not same, the terms are not in geometric sequence.

b.

To determine

To state: whether the terms $\frac{a_1}{b_1},\frac{a_2}{b_2},\frac{a_3}{b_3}$ form a geometric sequence or not.

Answer to Problem 54E

The terms $\frac{a_1}{b_1},\frac{a_2}{b_2},\frac{a_3}{b_3}$ form a geometric sequence.

Explanation of Solution

Since,

  $\begin{array}{l}{a}_1=2,a_2=6,a_3=18\\ b_1=1,b_2=5,b_3=25\end{array}$

Then, $\frac{a_1}{b_1}=\frac{2}{1},\frac{a_2}{b_2}=\frac{6}{5},\frac{a_3}{b_3}=\frac{18}{25}$

Here common ratio(r) is $\frac{6}{5}÷2=\frac{18}{25}÷\frac{6}{5}\Rightarrow \frac{3}{5}=\frac{3}{5}$ same

Hence this is a geometric sequence whose common ratio(r) $=\frac{3}{5}$

This common ratio is $\frac{1}{5}$ times common ratio of geometric sequence given by $a_n=2{(3)}^{n-1}$

Again this common ratio is $\frac{3}{25}$ times common ratio of geometric sequence given by $b_n=5^{n-1}$