Question

Divide 32 into 4 parts which are in AP such that ratio of product of extremes to the product of means is 7 is to 15

Answer 1

Answer:Let four numbers in A.P. be a – 3d, a – d, a + d and a + 3d.

(a – 3d) + (a – d) + (a + d) + (a + 3d) = 32

⇒ 4a = 32

⇒ a = 8

Product of extremes = (a – 3d) (a + d) + (8 – 3d) (8 + 3d) = 64 – 9d2

Product of means = (a – d) (a + d) = (8 – d) (8 + d) = 64 – d2

Product of extremes: product of means = 7:15  (Given)

When a = 8 and d = 2

a – 3d = 8 – 3 × 2 = 8 – 6 = 2

a – d = 8 – 2 = 6

a + d = 8 + 2 = 10

a + 3d = 8 + 3 × 2 = 8 + 6 = 14

When a = 8, d = –2

a – 3d = 8 – 3 × (–2) = 8 + 6 = 14

a – d = 8 – (–2) = 8 + 2 = 10

a + d = 8 – 2 = 6

a + 3d = 8 + 3 × (–2) = 8 – 6 = 2

∴ The four numbers are 2, 6, 10 and 14.

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