Finding the Difference Between Averages

This solution explains how to find the difference between the average of the first 50 even natural numbers and the average of the first 50 odd natural numbers.

Calculating the Average of First 50 Even Natural Numbers

The first 50 even natural numbers are:

2, 4, 6, 8, ..., up to the 50th even number.

The sequence of the first \(n\) even natural numbers is \(2, 4, 6, \dots, 2n\). In this case, \(n=50\), so the 50th even number is \(2 \times 50 = 100\). The sequence is \(2, 4, 6, \dots, 100\).

To find the average of an arithmetic sequence, we can use the formula:

\( \text{Average} = \frac{\text{First Term} + \text{Last Term}}{2} \)

Applying this formula:

\( \text{Average}_{\text{even}} = \frac{2 + 100}{2} = \frac{102}{2} = 51 \)

Alternatively, the average of the first \(n\) even natural numbers is given by the formula \(n+1\). For \(n=50\), the average is \(50 + 1 = 51\).

Calculating the Average of First 50 Odd Natural Numbers

The first 50 odd natural numbers are:

1, 3, 5, 7, ..., up to the 50th odd number.

The sequence of the first \(n\) odd natural numbers is \(1, 3, 5, \dots, 2n-1\). In this case, \(n=50\), so the 50th odd number is \(2 \times 50 - 1 = 100 - 1 = 99\). The sequence is \(1, 3, 5, \dots, 99\).

Using the same average formula for an arithmetic sequence:

\( \text{Average}_{\text{odd}} = \frac{\text{First Term} + \text{Last Term}}{2} \)

Applying this formula:

\( \text{Average}_{\text{odd}} = \frac{1 + 99}{2} = \frac{100}{2} = 50 \)

Alternatively, the average of the first \(n\) odd natural numbers is simply \(n\). For \(n=50\), the average is \(50\).

Calculating the Difference

The question asks for the difference between the average of the first 50 even natural numbers and the average of the first 50 odd natural numbers.

\( \text{Difference} = \text{Average}_{\text{even}} - \text{Average}_{\text{odd}} \)

\( \text{Difference} = 51 - 50 \)

\( \text{Difference} = 1 \)

Conclusion

The difference between the average of the first 50 even natural numbers and the average of the first 50 odd natural numbers is 1.