To find the average of all odd numbers up to 100, we first need to identify all the odd numbers in that range. The odd numbers from 1 to 99 are: 1, 3, 5, …, 99.
We can see that this is an arithmetic series with the first term (a) = 1, the last term (l) = 99, and the common difference (d) = 2. To find the number of terms (n), we can use the formula:
n = (l – a) / d + 1
Substituting in our values:
n = (99 – 1) / 2 + 1 = 49 + 1 = 50
Now that we know there are 50 odd numbers, we can find the sum (S) of the series using the formula:
S = n / 2 * (a + l)
Plugging in the values:
S = 50 / 2 * (1 + 99) = 25 * 100 = 2500
Now, to find the average (A), we divide the sum by the number of terms:
A = S / n = 2500 / 50 = 50
Therefore, the average of all odd numbers up to 100 is 50.