What is the weighted mean of first 10 natural numbers whose weights are equal to the corresponding number ?

Question

Accepted Answer

To determine the weighted mean of the first 10 natural numbers where the weights are equal to the corresponding numbers, we need to follow these steps:

1. **List the first 10 natural numbers**: These are 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.

2. **Assign weights equal to the numbers themselves**: This means the weight for 1 is 1, for 2 is 2, and so on up to 10.

3. **Calculate the weighted mean**: The formula for the weighted mean is given by:

$$
   \text{Weighted Mean} = \frac{\sum (x_i \cdot w_i)}{\sum w_i}
   $$

where $x_i$ is each natural number and $w_i$ is the corresponding weight.

4. **Calculate the numerator**: This is the sum of each number multiplied by its weight:

$$
   \sum (x_i \cdot w_i) = 1 \cdot 1 + 2 \cdot 2 + 3 \cdot 3 + \ldots + 10 \cdot 10
   $$

$$
   = 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 = 385
   $$

5. **Calculate the denominator**: This is the sum of the weights:

$$
   \sum w_i = 1 + 2 + 3 + \ldots + 10 = 55
   $$

6. **Compute the weighted mean**:

$$
   \text{Weighted Mean} = \frac{385}{55} = 7
   $$

Therefore, the weighted mean of the first 10 natural numbers with weights equal to the numbers themselves is 7, which corresponds to Option 1.

**Explanation for Incorrect Options:**

- **Option 2 (5.5)**: This value might be confused with the arithmetic mean of the first 10 natural numbers, which is indeed 5.5. However, the question specifically asks for the weighted mean, not the arithmetic mean.

- **Option 3 (5)**: This value does not correspond to any logical calculation related to either the arithmetic or weighted mean of the first 10 natural numbers.

- **Option 4 (4.5)**: Similar to Option 3, this value does not align with any calculation for the mean of the first 10 natural numbers, whether weighted or arithmetic.

In conclusion, the correct answer is Option 1, as the weighted mean calculation clearly results in 7.