Chapter 12, Problem 33RE

To determine

Show that $a_1{b}_1,a_2{b}_2,a_3{b}_3,\mathrm{.....}$ is also in geometric sequence if $a_1,a_2,a_3,\mathrm{.....}$ and $b_1,b_2,b_3,\mathrm{.....}$ are in geometric sequence

Answer to Problem 33RE

  $a_1{b}_1,a_2{b}_2,a_3{b}_3,\mathrm{.....}$ is also an geometric sequence

Explanation of Solution

Given information:

It is given in the question that, $a_1,a_2,a_3,\mathrm{.....}$ and $b_1,b_2,b_3,\mathrm{.....}$ are in geometric sequence.

Formula used:

Common ratio, $r\Rightarrow \frac{a_2}{a_1}=\frac{a_3}{a_2}$

Where,

  $a_{1}isthe1stterm$

  $a_{2}isthe2ndterm$

  $a_{3}isthe3rdterm$

Proof:

If $a_1,a_2,a_3,\mathrm{.....}$ and $b_1,b_2,b_3,\mathrm{.....}$ are in geometric sequence means the common ratio of two consecutive term are same for both given geometric sequence that is

  $r\Rightarrow \frac{a_2}{a_1}=\frac{a_3}{a_2}$

  $\frac{a_2}{a_1}=\frac{a_3}{a_2}\mathrm{.......}(a)$

And,

  $r\Rightarrow \frac{b_2}{b_1}=\frac{b_3}{b_2}$

  $\frac{b_2}{b_1}=\frac{b_3}{b_2}\mathrm{.......}(b)$

Form the another given equation $a_1{b}_1,a_2{b}_2,a_3{b}_3,\mathrm{.....}$

  $r\Rightarrow \left(\frac{a_2}{a_1}=\frac{a_3}{a_2}\right)$

  $\begin{array}{l}\Rightarrow \left(\frac{a_2{b}_2}{a_1{b}_1}=\frac{a_3{b}_3}{a_2{b}_2}\right)\\ \end{array}$

  $\begin{array}{l}\Rightarrow \left(\frac{a_2}{a_1}*\frac{b_2}{b_1}=\frac{a_3}{a_2}*\frac{b_3}{b_2}\right)\\ \end{array}$

By using the equation $\frac{a_2}{a_1}=\frac{a_3}{a_2}\mathrm{.......}(a)$ and $\frac{b_2}{b_1}=\frac{b_3}{b_2}\mathrm{.......}(b)$

  $\Rightarrow \left(\frac{a_3}{a_2}*\frac{b_3}{b_2}=\frac{a_3}{a_2}*\frac{b_3}{b_2}\right)$

Here, the above equation is true so $a_1{b}_1,a_2{b}_2,a_3{b}_3,\mathrm{.....}$ is also in geometric sequence