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 \(\frac{n(n + 1)}{2}\).
Assertion (A) : The mean of first 'n' natural numbers is \(\frac{n - 1}{2}\).
Reason (R) : The sum of first 'n' natural numbers is \(\frac{n(n + 1)}{2}\).
Reason (R) : The sum of first 'n' natural numbers is \(\frac{n(n + 1)}{2}\).
Show Hint
To quickly check formulas for "first \(n\) natural numbers", test with a small value of \(n\), such as \(n = 3\) (numbers: 1, 2, 3):
- True mean = \(\frac{1 + 2 + 3}{3} = \frac{6}{3} = 2\).
- Formula from Assertion: \(\frac{3 - 1}{2} = 1\). (Since \(1 \neq 2\), the assertion is immediately disproved).
- Formula from Reason: \(\frac{3(4)}{2} = 6\). (Correct).
Testing with small numbers is an excellent technique to avoid formula-related traps!
- True mean = \(\frac{1 + 2 + 3}{3} = \frac{6}{3} = 2\).
- Formula from Assertion: \(\frac{3 - 1}{2} = 1\). (Since \(1 \neq 2\), the assertion is immediately disproved).
- Formula from Reason: \(\frac{3(4)}{2} = 6\). (Correct).
Testing with small numbers is an excellent technique to avoid formula-related traps!
Updated On: Jul 7, 2026
- 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, but Reason (R) is not correct explanation of Assertion (A).
- Assertion (A) is true, but Reason (R) is false.
- Assertion (A) is false, but Reason (R) is true.
Show Solution
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Question:
This is an Assertion-Reason question. We need to evaluate the truth of Assertion (A) and Reason (R) individually and then select the correct option.
Step 2: Key Formula or Approach:
1. The sum of the first \(n\) natural numbers is given by:
\[ S_n = \frac{n(n + 1)}{2} \]
2. The mean of a set of \(n\) numbers is:
\[ \text{Mean} = \frac{\text{Sum of numbers}}{n} \]
Step 3: Detailed Explanation:
1.
Evaluate Reason (R):
The sum of the first \(n\) natural numbers \(1 + 2 + 3 + \dots + n\) is a standard AP sum:
\[ S_n = \frac{n(n + 1)}{2} \]
This formula is mathematically correct and true. Therefore, Reason (R) is true.
2.
Evaluate Assertion (A):
Using the sum formula from the Reason, we can calculate the mean of the first \(n\) natural numbers:
\[ \text{Mean} = \frac{\text{Sum}}{n} \]
\[ \text{Mean} = \frac{\frac{n(n + 1)}{2}}{n} \]
\[ \text{Mean} = \frac{n + 1}{2} \]
However, the Assertion states that the mean of the first \(n\) natural numbers is \(\frac{n - 1}{2}\).
Since \(\frac{n + 1}{2} \neq \frac{n - 1}{2}\), the Assertion (A) is false.
3. Combining our results: Assertion (A) is false, but Reason (R) is true. This corresponds to option (D).
Step 4: Final Answer:
Assertion (A) is false, but Reason (R) is true, which corresponds to option (D).
This is an Assertion-Reason question. We need to evaluate the truth of Assertion (A) and Reason (R) individually and then select the correct option.
Step 2: Key Formula or Approach:
1. The sum of the first \(n\) natural numbers is given by:
\[ S_n = \frac{n(n + 1)}{2} \]
2. The mean of a set of \(n\) numbers is:
\[ \text{Mean} = \frac{\text{Sum of numbers}}{n} \]
Step 3: Detailed Explanation:
1.
Evaluate Reason (R):
The sum of the first \(n\) natural numbers \(1 + 2 + 3 + \dots + n\) is a standard AP sum:
\[ S_n = \frac{n(n + 1)}{2} \]
This formula is mathematically correct and true. Therefore, Reason (R) is true.
2.
Evaluate Assertion (A):
Using the sum formula from the Reason, we can calculate the mean of the first \(n\) natural numbers:
\[ \text{Mean} = \frac{\text{Sum}}{n} \]
\[ \text{Mean} = \frac{\frac{n(n + 1)}{2}}{n} \]
\[ \text{Mean} = \frac{n + 1}{2} \]
However, the Assertion states that the mean of the first \(n\) natural numbers is \(\frac{n - 1}{2}\).
Since \(\frac{n + 1}{2} \neq \frac{n - 1}{2}\), the Assertion (A) is false.
3. Combining our results: Assertion (A) is false, but Reason (R) is true. This corresponds to option (D).
Step 4: Final Answer:
Assertion (A) is false, but Reason (R) is true, which corresponds to option (D).
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