The product of three numbers in A.P. is 224, and the largest number is 7 times the smallest. Find the smallest number.

Sure, let's solve this step by step.

Given that the three numbers are in Arithmetic Progression (A.P.), they can be represented as a-d, a, a+d where 'a' is the middle term and 'd' is the common difference.

Also, we know that the product of the three numbers is 224. So, we can write the equation as:

(a-d) * a * (a+d) = 224

Simplifying this, we get:

a^3 - d^2 * a = 224 ---- (1)

We are also given that the largest number is 7 times the smallest. So, we can write the equation as:

a + d = 7 * (a - d)

Solving this, we get:

8d = 6a

So, a = 4/3 * d ---- (2)

Substituting equation (2) in equation (1), we get:

(4/3 * d)^3 - d^2 * (4/3 * d) = 224

Solving this, we get:

d = 3

Substituting d = 3 in equation (2), we get:

a = 4

So, the smallest number (a - d) is 4 - 3 = 1.