PHP program to find the average of the first n natural numbers that are even

To find the average of the first n natural numbers that are even, we need to calculate the sum of the first n even natural numbers and divide by n. The first n even natural numbers are 2, 4, 6, 8, ..., 2n.

Mathematical Formula

The average of the first n even natural numbers can be calculated using the formula: Average = (n + 1). This is because the sum of first n even numbers is n(n+1) and dividing by n gives us (n+1).

Example

<?php
function even_nums_avg($n)
{
    return $n + 1;
}

$n = 11;
echo "The average of the first " . $n . " natural numbers that are even is: ";
echo even_nums_avg($n);
?>

Output

The average of the first 11 natural numbers that are even is: 12

How It Works

The function 'even_nums_avg' uses the mathematical formula where the average of first n even natural numbers equals (n + 1). For n = 11, the first 11 even numbers are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22. Their sum is 132, and 132 ÷ 11 = 12, which matches our formula result.

Alternative Approach

Here's another implementation that calculates the sum explicitly ?

<?php
function even_nums_avg_manual($n)
{
    $sum = 0;
    for ($i = 1; $i <= $n; $i++) {
        $sum += 2 * $i;  // 2*i gives us the i-th even number
    }
    return $sum / $n;
}

$n = 5;
echo "Using manual calculation for n = " . $n . ": ";
echo even_nums_avg_manual($n);
?>

Using manual calculation for n = 5: 6

Conclusion

The average of the first n even natural numbers is simply (n + 1). This mathematical insight allows for an efficient O(1) solution instead of calculating the sum iteratively.