Chapter 12, Problem 34RE

a.

To determine

If $a_1,a_2,a_3,\mathrm{......}$ is an arithmetic sequence then find this sequence $a_1+2,a_2+2,a_3+2,\mathrm{......}$ is also arithmetic or not

Answer to Problem 34RE

  $a_1+2,a_2+2,a_3+2,\mathrm{......}$ is also an arithmetic sequence.

Explanation of Solution

Given information:

It is given in the question that, $a_1,a_2,a_3,\mathrm{......}$ is an arithmetic sequence

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:

  $a_1,a_2,a_3,\mathrm{......}$ is an arithmetic sequence means,

  $d\Rightarrow a_2-a_1=a_3-a_2$

And, in $a_1+2,a_2+2,a_3+2,\mathrm{......}$

Also $d\Rightarrow a_2-a_1=a_3-a_2$

It is clear from the above that,

Here, $a_1+2,a_2+2,a_3+2,\mathrm{......}$ is also an arithmetic sequence.

b.

To determine

If $a_1,a_2,a_3,\mathrm{......}$ is geometric sequence then find this sequence $5a_1,5a_2,5a_3,\mathrm{......}$ is also in geometric or not

Answer to Problem 34RE

  $5a_1,5a_2,5a_3,\mathrm{......}$ is also in geometric sequence.

Explanation of Solution

Given information:

It is given in the question that, $a_1,a_2,a_3,\mathrm{......}$ is geometric sequence.

Formula Used:

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

Where,

  $a_{1}isthe1stterm$

  $a_{2}isthe2ndterm$

  $a_{3}isthe3rdterm$

Calculation:

  $a_1,a_2,a_3,\mathrm{......}$ is geometric sequence means,

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

And in $5a_1,5a_2,5a_3,\mathrm{......}$ geometric sequence also the common ratio

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

Here, the above equation is true so $5a_1,5a_2,5a_3,\mathrm{......}$ is also in geometric sequence