Question

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

Answer 1

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