Question

divide 32 into 4 parts in AP such that product of exremes is to product of means is 7:15

Answer 1

Hi!
Here is the answer to your question.

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.

Cheers!

Answer 2

Let the our parts are a-3d,a-d,a+d and a+3d.
Then by the given question, 
a-3d+a-d+a+d+a+3d=32
or, 4a=32
or, a=8 and
(a-3d)(a+3d)/(a-d)(a+d)=7/15
or, a²-9d²/a²-d²=7/15
or, 15a²-135d²=7a²-7d²
or, 15a²-7a²=-7d²+135d²
or, 8a²=128d²
or, 8×8²=128d²
or, d²=8×8×8/128
or, d²=4
or, d=+-2
∴, The four parts are:
when d=2
8-(3×2)=8-6=2
8-2=6
8+2=10
8+(3×2)=14
when d=-2
8-(3×-2)=8+6=14
8-(-2)=8+2=10
8+(-2)=8-2=6
8+(3×-2)=8-6=2


Hope This helps :)