Answer :

Let 32 be divided into parts as (a - 3d), (a - d), (a + d) and (a + 3d).

Now (a - 3d) + (a - d) + (a + d) + (a + 3d) = 32

⇒ 4a = 32

⇒ a = 8

Now, we are given that product of the first and the fourth terms is to the product of the second and the third terms as 7 : 15.

i.e. [(a - 3d) × (a + 3 d)] : [(a - d) × (a + d)] = 7 : 15

⇒ =

⇒ 15[(a - 3d) × (a + 3 d)] = 7[(a - d) × (a + d)]

⇒ 15[a^{2} - 9d^{2}] = 7 [a^{2} - d^{2}]

⇒ 15a^{2} - 135d^{2} = 7a^{2} - 7d^{2}

⇒ 8a^{2} - 128d^{2} = 0

⇒ 8a^{2} = 128d^{2}

Putting the value of a, we get,

512 = 128 d^{2}

⇒ d^{2} = 4

⇒ d = ±2

∴ If d = 2, then the numbers are 2, 6, 10, 14.

If d = - 2, then the numbers are 14, 10, 6, 2.

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