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

To solve the problem, let's denote the three numbers in arithmetic progression (A.P.) as:

• a - d (smallest number)
• a (middle number)
• a + d (largest number)

According to the problem, the product of these three numbers is 224:

(a - d) × a × (a + d) = 224

We also know that the largest number is 7 times the smallest:

a + d = 7(a - d)

Now, let's simplify the second equation:

a + d = 7a - 7d

Rearranging gives:

8d = 6a

From this, we can express d in terms of a:

d = (3/4)a

Substituting this value of d back into the product equation:

(a - (3/4)a) × a × (a + (3/4)a = 224

This simplifies to:

(1/4)a × a × (7/4)a = 224

Which further simplifies to:

(7/16)a^3 = 224

Now, solving for a:

a^3 = 224 × (16/7)

a^3 = 512

a = 8

Now, substituting back to find d:

d = (3/4) × 8 = 6

Now we can find the three numbers:

• a - d = 8 - 6 = 2
• a = 8
• a + d = 8 + 6 = 14

Thus, the three numbers are 2, 8, and 14.