Question

Assertion (A): a, b, c are in A.P. if and only if 2b = a + c.

Reason (R): The sum of first n odd natural numbers is n².

Options

• Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

• Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).

• Assertion (A) is true but Reason (R) is false.

• Assertion (A) is false but Reason (R) is true.

Answer 1

Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).

Explanation:

If a, b, c are in A.P.,

Then c – b = b – a

`\implies` c + a = 2b

`\implies` 2b = a + c

∴ Assertion (A) is true.

Now, first n odd natural numbers are

1, 3, 5, 7, 9, ............. (2n – 1)

Here, First term (a) = 1

Common difference (d) = 3 – 1 = 2

Last term (a_n) = 2n – 1

The sum of an A.P. series

S = `n/2[2a + (n - 1)d]`

= `n/2[2 xx 1 + (n - 1) xx 2]`

= `n/2[2 + 2n - 2]`

= `n/2 xx 2n`

= n²