CBSE Class 10 Maths Question Paper 2018 Solution | Q24

Question 24: The sum of four consecutive numbers in an AP is 32 and the ratio of the product of the first and the last term to the product of two middle terms is 7 : 15. Find the numbers.

Video Explanation

Explanatory Answer | CBSE Class 10 Math Question Paper 2018 Q24

Let the 4 numbers in AP be x - 3d, x - d, x + d and x + 3d, where 2d is the common difference
Given Data: Sum of the 4 numbers is 32.
Sum of 4 numbers = (x - 3d) + (x - d) + (x + d) + (x + 3d) = 4x
4x = 32 or x = 8
∴ The 4 numbers are = (8 - 3d), (8 - d), (8 + d), and (8 + 3d)

Product of first and last term = (8 - 3d) (8 + 3d) = 64 – 9d²
Product of the 2 middle terms = (8 - d) (8 + d) = 64 – d²

Given Data: The ratio of the product of the first and the last term to the product of two middle terms is 7 : 15
\\frac{64 - 9d^2}{64 - d^2}) = \\frac{7}{15})
15(64 – 9d²) = 7(64 - d²)
15 × 64 - 135d² = 7 × 64 - 7d²
8 × 64 = 128d²
d² = \\frac{8 × 64}{128}) = 4
∴ d = ± 2

If d = 2, the terms are (8 – 3 × 2) + (8 - 2) + (8 + 2) + (8 + 3 × 2)
2, 6, 10, 14.

If d = -2, the terms are (8 – 3 × (-2)) + (8 - (-2)) + (8 + (-2)) + (8 + 3 × (-2))
14, 10, 6, 2.