Let our terms are like this:-

a-3d, a-d, a+d, a+3d

given that the sum of its all terms is 32.
i.e.
a-3d+a-d+a+d+a+3d=32
4a=32
[a=8]

now, according to second condition product of first and last term with respect to product of two middle terms is in ratio of 7:15.

(a-3d)(a+3d)/(a-d)(a+d)=7/15
a^2-9d^2/a^2-d^2=7/15
64-9d^2/64-d^2=7/15
448-7d^2=960-135d^2.------[by cross multiplication]
128d^2=512
d^2=512/128
d=√4
[d=2]

So, now we have:-
a=8 and d=2

hence the terms will be:-
A1= a-3d = 8-3×2 = 2
A2= a-d = 8-2 = 6
A3= a+d = 8+2 = 10
A4= a+3d = 8+3×2 = 14

So,
AP= 2, 6, 10, 14

Thank you.. :-)

Problem Statement:
The sum of four consecutive numbers in an Appointment is 32 and the ratio of the product of the first and the last terms to the product of the two middle terms is 7:15. Find the numbers.

Solution:
Let's assume the four consecutive numbers as x, x+1, x+2, and x+3.

Step 1: Set up the Equations
We are given two conditions:
1. The sum of the four consecutive numbers is 32.
2. The ratio of the product of the first and the last terms to the product of the two middle terms is 7:15.

Step 2: Translate the Conditions into Equations
1. The sum of the four consecutive numbers is 32:
x + (x+1) + (x+2) + (x+3) = 32

2. The ratio of the product of the first and the last terms to the product of the two middle terms is 7:15:
(x * (x+3)) / ((x+1) * (x+2)) = 7/15

Step 3: Solve the Equations
We can solve the equations using algebraic methods.

1. The sum of the four consecutive numbers is 32:
x + (x+1) + (x+2) + (x+3) = 32
4x + 6 = 32
4x = 26
x = 6.5

2. The ratio of the product of the first and the last terms to the product of the two middle terms is 7:15:
(x * (x+3)) / ((x+1) * (x+2)) = 7/15
(6.5 * (6.5+3)) / ((6.5+1) * (6.5+2)) = 7/15
(6.5 * 9.5) / (7.5 * 8.5) = 7/15
61.75 / 63.75 = 7/15

Step 4: Check the Solution
We can check if the solution is correct by substituting the value of x back into the equations.

1. The sum of the four consecutive numbers is 32:
6.5 + (6.5+1) + (6.5+2) + (6.5+3) = 32
6.5 + 7.5 + 8.5 + 9.5 = 32
32 = 32 (True)

2. The ratio of the product of the first and the last terms to the product of the two middle terms is 7:15:
(6.5 * (6.5+3)) / ((6.5+1) * (6.5+2)) = 7/15
(6.5 * 9.5) / (7.5 * 8.5) = 7/15
61.75 / 63.75 = 7/15
0.9686 ≈ 0.9333