The n^th term of a sequence = 5n² - 3.

Statement (1): The sequence is an A.P.

Statement (2): If the n^th term of a sequence is not linear, the sequence does not form an A.P.

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

Given, a_n = 5n² - 3

⇒ a₁ = 5(1)² - 3 = 5 - 3 = 2

⇒ a₂ = 5(2)² - 3 = 5 × 4 - 3 = 20 - 3 = 17

⇒ a₃ = 5(3)² - 3 = 5 × 9 - 3 = 45 - 3 = 42.

Difference between terms :

⇒ a₃ - a₂ = 42 - 17 = 25

⇒ a₂ - a₁ = 17 - 2 = 15

Since, the difference between consecutive terms is not equal. Thus, the sequence is not in an A.P.

So, statement 1 is false.

The general form of an A.P. is a_n = a + (n - 1)d, which is a linear expression in n.

So, if T_n is not linear in n, then the sequence cannot be an A.P.

So, statement 2 is true.

Hence, option 4 is the correct option.