Question

What is the common difference of four terms in AP. Such that the ratio of product of first fourth term to that of the second and third term is 2:3 and the sum of all terms is 20

Answer 1

Let four terms ; (a - 3d), (a - d) , (a + d), (a + 3d) are in AP.

A.T.Q

Sum of all four terms is 20

(a - 3d) + (a - d) + (a + d) + (a + 3d) = 20

4a = 20

a = 5 .....................(1)

And

The ratio of the product of first and fourth terms to that of second term and third is 2:3.

(a - 3d)(a + 3d)/(a - d)(a + d) = 2/3

(a² - 9d²)/(a² - d²) = 2/3

3a² - 27d² = 2a² - 2d²

a² = 25d²

(5)² = 25d². [ from (1),]

d² = 1

d = ±1

common difference of four terms in AP is

= (a - d) - (a - 3d) = (a + d) - (a - d) = (a + 3d) - (a + d) = 2d

= 2(±1)

= ±2