the mean of first n natural number is 61 then n value is 

To find the value of \( n \) when the mean of the first \( n \) natural numbers is 61, we can set up the equation as follows:

### Step-by-Step Solution

1. **Mean of First \( n \) Natural Numbers:**
   The mean of the first \( n \) natural numbers is given by:
   \[
   \text{Mean} = \frac{\text{Sum of first } n \text{ natural numbers}}{n}
   \]

2. **Sum of First \( n \) Natural Numbers:**
   The sum of the first \( n \) natural numbers is:
   \[
   \text{Sum} = \frac{n(n+1)}{2}
   \]

3. **Setting up the Mean Equation:**
   Since the mean is given as 61, we have:
   \[
   \frac{\frac{n(n+1)}{2}}{n} = 61
   \]

4. **Simplify the Equation:**
   Simplifying the left side, we get:
   \[
   \frac{n(n+1)}{2n} = 61
   \]
   \[
   \frac{n+1}{2} = 61
   \]

5. **Solve for \( n \):**
   Multiply both sides by 2:
   \[
   n + 1 = 122
   \]
   Subtract 1 from both sides:
   \[
   n = 121
   \]

### Answer
The value of \( n \) is **121**.

Would you like further details on the steps or have additional questions?

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Here are five related questions for further exploration:

1. What is the sum of the first 121 natural numbers?
2. How would the mean change if \( n \) were doubled?
3. How can you find the mean of the first \( n \) odd numbers?
4. How does the formula for the sum of squares of the first \( n \) natural numbers work?
5. What would the mean be if \( n = 100 \) instead?

**Tip:** Remember, the mean of the first \( n \) natural numbers always increases as \( n \) increases since each subsequent number adds a positive value to the average.

the mean of first n natural number is 61 then n value is

Learn how to find the value of n when the mean of the first n natural numbers is 61. This problem uses arithmetic mean and the formula for the sum of natural numbers, ideal for students in Grades 8-10.

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