Arithmetic Progression (A.P.)
An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference.

Geometric Progression (G.P.)
A geometric progression is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

Given Information
- The numbers a, X, c are in A.P. with X = 25.
- The numbers a, Y, c are in G.P. with Y = 7.

Step 1: Find the Common Difference (d) of A.P.
Since a, X, c are in A.P., we can write:
X = a + d
c = X + d

Substituting the value of X, we get:
c = (a + d) + d
c = a + 2d

Step 2: Find the Common Ratio (r) of G.P.
Since a, Y, c are in G.P., we can write:
Y = a * r
c = Y * r

Substituting the value of Y, we get:
c = (a * r) * r
c = a * r^2

Step 3: Equate the values of c
From Step 1, we have c = a + 2d.
From Step 2, we have c = a * r^2.

Equate the two expressions for c:
a + 2d = a * r^2

Step 4: Solve the Equation
We have two unknowns, a and d, and one equation. To solve for these variables, we need another equation.

Step 5: Use the Given Information
We know that X = 25 and Y = 7.

From Step 1, we have X = a + d.
Substituting the value of X, we get:
25 = a + d

From Step 2, we have Y = a * r.
Substituting the value of Y, we get:
7 = a * r

Step 6: Solve the Second Equation
We can rearrange the equation 7 = a * r to solve for r:
r = 7 / a

Step 7: Substitute the Values of r and X
Substitute the value of r = 7 / a into the equation a + 2d = a * r^2:
a + 2d = a * (7 / a)^2
a + 2d = a * (49 / a^2)
a + 2d = 49 / a

Step 8: Solve for a
Cross-multiplying the equation:
a(a + 2d) = 49
a^2 + 2ad = 49

Step 9: Substitute the Value of X
Substitute the value of X = 25 into the equation a + d = 25:
a + d = 25
d = 25 - a

Step 10: Solve the System of Equations
Substitute the value of