Chapter 12, Problem 35RE

a.

To determine

Calculate the value of x for which the sequence $6,x,12,\mathrm{.....}$ is arithmetic

Answer to Problem 35RE

The value of ‘x’ must be $x=9$ for which the sequence $6,x,12,\mathrm{.....}$ is arithmetic

Explanation of Solution

Given information:

It is given in the question that, the sequence $6,x,12,\mathrm{.....}$

Formula Used:

Common difference, $d\Rightarrow a_2-a_1=a_3-a_2$

Where,

  $a_{1}isthe1stterm$

  $a_{2}isthe2ndterm$

  $a_{3}isthe3rdterm$

Calculation:

Sequence $6,x,12,\mathrm{.....}$ is arithmetic sequence means the common difference of two consecutive term must be same.

  $d\Rightarrow a_2-a_1=a_3-a_2$

  $\begin{array}{l}\Rightarrow x-6=12-x\\ \Rightarrow 2x=18\\ \Rightarrow x=9\end{array}$

It is clear from the above solution that $x=9$ for which the sequence $6,x,12,\mathrm{.....}$ is arithmetic

b.

To determine

Calculate the value of x for which the sequence $6,x,12,\mathrm{.....}$ is geometric

Answer to Problem 35RE

The value of ‘x’ must be $x=\sqrt{72}$ for which the sequence $6,x,12,\mathrm{.....}$ is geometric

Explanation of Solution

Given information:

It is given in the question that, the sequence $6,x,12,\mathrm{.....}$

Formula Used:

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

Where,

  $a_{1}isthe1stterm$

  $a_{2}isthe2ndterm$

  $a_{3}isthe3rdterm$

Calculation:

Sequence $6,x,12,\mathrm{.....}$ is geometric sequence means the common ratio of two consecutive term must be same.

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

  $\begin{array}{l}\frac{x}{6}=\frac{12}{x}\\ x^2=72\\ x=\sqrt{72}\end{array}$

It is clear from the above solution that $x=\sqrt{72}$ for which the sequence $6,x,12,\mathrm{.....}$ is arithmetic