Question

Find the average of the first 20 multiples of 7.

Answer 1

Hint: A multiple of a number can be defined as a number which is completely divisible by original number. So, we must find the first twenty multiples of seven and then try to formulate series using these multiples. Once a definite series is formed, then we can apply certain useful formulas to calculate the average of the first 20 multiples of 7.

Complete step-by-step answer:
In simple mathematics if we multiply two numbers a and b, then b is the multiple of a. So, the meaning of multiple is the product result of one number multiplied by another number.
For example, 12 is a multiple of 3 and 4 as the product of 3 and 4 yields 12.
In our problem we are provided the number 7. We are required to calculate the average of the first 20 multiples of 7.
So, multiples of 7 are: $7\times 1,7\times 2,7\times 3,7\times 4....................\times 7\times 20$.
By closely observing the variables of multiples of 7 we try to formulate a series.
Now, a recurring pattern could be developed by taking 7 common from the multiples.
Taking 7 common from multiples we get,
$7\cdot \left( 1+2+3+4...................+19+20 \right)$
As we have obtained a series for multiples of 7, now we can apply the sum of a series and evaluate the total sum of the first 20 multiples of 7.
By using series formula, we get ${{S}_{n}}=\dfrac{n(n+1)}{2}$
So, for the first 20 multiples the number of terms for our case is 20.
Therefore, sum would be: $\dfrac{7\cdot \left( 20 \right)\cdot \left( 20+1 \right)}{2}=1470$
Now, to find the average we divide sum by total number of terms i.e. 20.
$A=\dfrac{1470}{20}=73.5$
Hence, the average of the first 20 multiples of 7 is 73.5.

Note: The key step is to form a series and apply summation of a series to reduce the amount of calculations. The knowledge of the formula for summation of a series makes the problem easy to calculate. Manual calculation of the sum becomes very lengthy and will also increase the chances of error.