To find the average arithmetic mean of all the multiples of ten from 10 to 190, first, we need to identify the multiples of ten within this range: 10, 20, 30, …, 190.
These numbers form an arithmetic series where:
- The first term (a) is 10.
- The last term (l) is 190.
- The common difference (d) is 10.
Next, we can determine how many terms (n) are in this series. We can use the formula for the nth term of an arithmetic sequence:
l = a + (n – 1) * d
Plugging in the values gives us:
190 = 10 + (n – 1) * 10
Simplifying this, we find:
190 – 10 = (n – 1) * 10
180 = (n – 1) * 10
n – 1 = 18
n = 19
Now, we have 19 terms. To find the average, we need the sum of these terms. The sum (S) of an arithmetic series can be calculated using the formula:
S = n/2 * (a + l)
Substituting our values:
S = 19/2 * (10 + 190)
S = 19/2 * 200
S = 19 * 100
S = 1900
Finally, we can calculate the average:
Average = S / n
Average = 1900 / 19
Average = 100
Therefore, the average arithmetic mean of all the multiples of ten from 10 to 190 is 100.