if divided 20 into four parts which are in ap such that product of the fourth is to the product of the second and third is the same as 2:3, then the smallest part is-

Let the numbers are a-3d,a-d,a+d and a+3d.

Now ratio to the foruth to the the product of the second  and third is the same as 2:3,

a+3d      =      2

(a+d)(a-3d)     3

Solve and you will get ans.

let the 4 terms of AP be : a-3d, a-d, a+d, a+3d.

then the sum of 4 terms is a-3d +a-d+a+d + a+3d = 4a

given that 4a= 20.

therefore, a=5.

now, again given that

(a+3d)/(a-d)(a+d) = 2/3

solving, we get d=-7, 5/2

now, least term is a-3d.

if d is+ve,then the value of term is least.

keeping the values for d and a, least term=-5/2

please approve my answer.

Let the four parts be (a-d), (a), (a+d), (a+2d).

Then, (a-d)+(a)+(a+d)+(a+2d)=20

2a+d=10

a=(10-d)/2

Now,

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