Chapter 11.3, Problem 40WE

To determine

To calculate: The value of x and y .

Answer to Problem 40WE

The value of x and y are $\frac{52}{3}\text{ and }\frac{80}{3}$ respectively.

Explanation of Solution

Given information: The sequence is $8,x,y,36$

Case - I

According to the question, first three numbers are in an arithmetic sequence.

Therefore, $8,x,y$

Let, first term, $a=8$

Second term, $a_1=x$

Third term, $a_2=y$

Common difference, $d=a_1-a=a_2-a_1$

  $\begin{array}{l}\Rightarrow x-8=y-x\\ \Rightarrow 2x-y=8\\ \Rightarrow y=2x-8\mathrm{...}\left(i\right)\end{array}$

Case − II

According to the question, first three numbers are in an arithmetic sequence.

Therefore, $x,y,36$

Let, first term, $a=x$

Second term, $a_1=y$

Third term, $a_2=36$

Common difference, $d=a_1-a=a_2-a_1$

  $\begin{array}{l}\Rightarrow y-x=36-y\\ \Rightarrow 2y-x=36\mathrm{...}\left(ii\right)\end{array}$

From equation (i) and equation (ii), we get

Calculation:

  $\begin{array}{l}\Rightarrow 2\left(2x-8\right)-x=36\\ \Rightarrow 4x-16-x=36\\ \Rightarrow 3x=36+16\\ \Rightarrow 3x=52\\ \Rightarrow x=\frac{52}{3}\end{array}$

Put the value of x in equation (i), we get

Calculation:

  $\begin{array}{l}\Rightarrow y=2\times \frac{52}{3}-8\\ \Rightarrow y=\frac{104-24}{3}=\frac{80}{3}\end{array}$

Hence, the value of x and y are $\frac{52}{3}\text{ and }\frac{80}{3}$ respectively.