Answer :

Let

Recall that the value of a determinant remains same if we apply the operation R_{i}→ R_{i} + kR_{j} or C_{i}→ C_{i} + kC_{j}.

Applying R_{1}→ R_{1} + R_{3}, we get

Given that a, b and c are in an A.P. Using the definition of an arithmetic progression, we have

b – a = c – b

⇒ b + b = c + a

⇒ 2b = c + a

∴ a + c = 2b

By substituting this in the above equation to find Δ, we get

Taking 2 common from R_{1}, we get

Applying R_{1}→ R_{1} – R_{2}, we get

∴ Δ = 0

Thus, when a, b and c are in A.P.

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