Question

What is the average of the first 20 multiples of 3?

Answer 1

Hint: A number that is totally divisible by the original number is referred to as a multiple. As a result, we must first identify the first twenty-seven-digit multiples before attempting to construct series utilizing these multiples. We can use some useful formulas to get the average of the first 20 multiples of 3 once we've established a clear series.


Complete step-by-step solution:
If we multiply two numbers, a and b, we get b, which is the multiple of a. As a result, the definition of multiple is the outcome of multiplying one integer by another. 12 is a multiple of 3 and 4 because the product of 3 and 4 equals 12.
We get the number 3 in our problem. We will find the average of the first 20 multiples of 3
So, the multiples of 3 are:
$3 \times 1,3 \times 2,3 \times 3,......,3 \times 19,3 \times 20.$
We can observe that each multiple of 3 contains a common term.
So, we take 3 common terms from each term. Then we get
$3 \times (1 + 2 + 3 + 4 + 5 + 6 + 7 + .... + 20)$
We can now use the sum of a series formula to calculate the entire sum of the first 20 multiples of 3 because we have a series for multiples of 3.
We know the series formula. i.e. ${S_n} = \dfrac{{n(n + 1)}}{2}$
So, for the first 20 multiples the number of terms for our case is 20.
Therefore, sum of first 20 multiples of 3 are
${S_{20}} = \dfrac{{3 \times 20 \times (20 + 1)}}{2}$
${S_{20}} = 630$
Now, to find the average we divide sum by total number of terms i.e. 20.
$ \Rightarrow A = \dfrac{{630}}{{20}} = 31.5$
Hence, the average of the first 20 multiples of 3 is 31.5.

Note: Forming a series and applying summation of a series to reduce the number of calculations is the most important stage. The problem is simple to calculate if you know the formula for summation of a series. The sum will be calculated manually, which will take a long time and increase the risks of error.