If the average of a₁, a₂, a₃ and a₄ is 19. 5, a₁ = 21 and the average of a₁, a₂ and a₃, is equal to the average of a₂, a₃ and a₄, then what will be the value of a₄           ?

Correct option is D

Given:

The average of $$a_1, a_2, a_3$$​, and $$a_4$$​ is 19.5, and $$a_1 = 21 .$$​​

It is also given that the average of $$a_1, a_2, \ and \ a_3$$​ is equal to the average of $$a_2,\ a_3, \ and \ a_4$$​. We need to find the value of $$a_4$$​ .

Concept Used

The average of a set of numbers is the sum of the numbers divided by the count of the numbers. Thus, for any set $$\{a_1, a_2, a_3, a_4\}$$​ with average A :
​$$A = \frac{a_1 + a_2 + a_3 + a_4}{4}$$​​

Solution

Using the information that the average of $$a_1, a_2, a_3,\ and\ a_4\ is\ 19.5:$$​​
​$$\frac{a_1 + a_2 + a_3 + a_4}{4} = 19.5$$​​

Multiply both sides by 4 to find the sum of $$a_1, a_2, a_3,\ and\ a_4$$​:

​$$a_1 + a_2 + a_3 + a_4 = 19.5 \times 4 = 78$$​​

Substitute $$a_1 = 21$$​ into the equation:
​$$21 + a_2 + a_3 + a_4 = 78$$​​

​$$a_2 + a_3 + a_4 = 57$$​​

Let the average of  $$a_1, a_2,\ and\ a_3$$​be equal to the average of $$a_2, a_3, \ and \ a_4$$​ .

Thus:
​$$\frac{a_1 + a_2 + a_3}{3} = \frac{a_2 + a_3 + a_4}{3}$$​​

Since a_1 = 21 , we conclude that a_4 = 21 .

The value of a_4 is 21.