Question

The n^th term of a sequence = `5 n^2 - 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.

Options

• Both the statements are true.

• Both the statements are false.

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

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

Answer 1

Statement 1 is false, and statement 2 is true.

Explanation:

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.