Question

Assertion (A) : The mean of first 'n' natural numbers is \( \frac{n - 1}{2} \).
Reason (R) : The sum of first 'n' natural numbers is \( \frac{n(n + 1)}{2} \).

Hint:

The mean of an A.P. is simply the average of the first and last terms: \( (1 + n)/2 \).

Answer 1

Assertion (A): The mean of the first \( n \) natural numbers is \[ \frac{n - 1}{2} \]

Reason (R): The sum of the first \( n \) natural numbers is \[ \frac{n(n + 1)}{2} \]

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Step 1: Check the Assertion
Mean of first \( n \) natural numbers:
\[ \text{Mean} = \frac{\text{Sum}}{\text{Number of terms}} = \frac{\frac{n(n+1)}{2}}{n} \] Simplify:
\[ = \frac{n+1}{2} \]
So the correct mean is \[ \boxed{\frac{n+1}{2}} \]
But the assertion says the mean is \[ \frac{n-1}{2} \] which is incorrect.

✔ Assertion (A) is false.

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Step 2: Check the Reason
The sum of the first \( n \) natural numbers is correctly given by:
\[ \frac{n(n+1)}{2} \]
✔ Reason (R) is true.

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Final Conclusion:
• Assertion (A) is false.
• Reason (R) is true.
• Since A is false and R is true, R does not explain A.

\[ \boxed{\text{A is false, R is true}} \]