Question

Assertion (A): The mean of first 'n' natural numbers is `(n - 1)/2`.

Reason (R): The sum of first 'n' natural numbers is `(n(n + 1))/2`.

Options

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

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

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

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

Answer 1

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

Explanation:

Assertion (A):

First n natural numbers are 1, 2, 3, ..., n.

Mean = `"Sum of observations"/"Number of observations"`

Mean = `((n(n + 1))/2)/n`

Mean = `(n + 1)/2` 

The assertion states the mean is `(n - 1)/2`, which is incorrect.

So, Assertion (A) is false.

Reason (R):

Sum of first n natural numbers = 1 + 2 + 3 + ... + n

Using A.P. sum formula:

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

`S_n = (n(n + 1))/2`

So, Reason (R) is true.