Answer:

Given that a, b, c are in A.P.

We know that in an A.P., the common difference is given by d = b – a = c – b.

Hence, we can write b = a + d and c = a + 2d.

Substituting these values in the expression, we get:

(a² + 4ac + c²)/(ab + bc + ca)

= (a² + 4a(a + 2d) + (a + 2d)²)/(a(a + d) + (a + d)(a + 2d) + a(a + 2d))

= (a² + 4a² + 8ad + a² + 4ad + 4d²)/(a² + ad + ad + 2ad + d²)

= (2a² + 12ad + 4d²)/(4ad + 2a² + d²)

= 2(a² + 6ad + 2d²)/(2ad + a² + d²)

= 2[(a + 2d)²]/[(a + d)²]

= 2(c/a)²

Therefore, the value of the expression is 2(c/a)².

Hence, we have derived the expression for the given question.

Summary:

Given expression: (a² 4ac c²)/(ab bc ca)

Value of expression: 2(c/a)²

Explanation:
- A.P. means a, b, c are in arithmetic progression
- Substituting b = a + d and c = a + 2d in the expression
- Simplifying the expression by expanding the numerator and denominator
- Finally, the expression is simplified to 2(c/a)².