Given x, y, z are in A.P. Assertion (A): z, y, x are in A.P. Reason (R): The term of an A.P. taken in reverse order also forms an A.P. 1. Assertion (A) is true, but Reason (R) is false. 2. Assertion (A) is false, but Reason (R) is true. 3. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A). 4. Both Assertion (A) and Reason (R) are correct, and Reason (R) is incorrect reason for Assertion (A).

Mathematics

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Given x, y, z are in A.P.

Common difference, d = difference of two consecutive terms,

⇒ a₂ - a₁ = y - x

⇒ a₃ - a₂ = z - y

If x, y, z are in A.P., then d will be equal, y - x = z - y

⇒ x - z = 2y ………………(1)

If z, y, x are in A.P.

⇒ a₂ - a₁ = y - z

⇒ a₃ - a₂ = x - y

If z, y, x are in A.P., then d will be equal, y - z = x - y

⇒ -z + x = 2y

⇒ z - x = -2y

From equation (1), we can write that z, y, x are in A.P. with negative d.

So, assertion (A) is true.

From above we can conclude that the term in A.P. taken in reverse order also form an A.P.

So, reason (R) is true. And, reason (R) is the correct explanation of assertion (A).

Thus, both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct reason for Assertion (A).

Hence, option 3 is the correct option.

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