Question

what is common difference of 4 terms of an ap such that the ratio of the product of the 1st and 4th terms to that of the 2nd and 3rd terms is 2:3and the sum of the 4 terms is 20​

Answer 1

Answer:

answer : 2 or -2

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

a/c to question,

sum of all four terms is 20

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

or, 4a = 20

or, a = 5......(1)

again a/c to question,

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

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

or, (a² - 9d²)/(a² - d²) = 2/3

or, 3a² - 27d² = 2a² - 2d²

or, a² = 25d²

from equation (1),

(5)² = 25d²

or, d² = 1 => d = ±1

here common difference = (a - d) - (a - 3d) = (a + d) - (a - d) = (a + 3d) - (a + d) = 2d

so, common difference = 2(±1) = ±2