Question

# Find the average of first 20 multiples of 7.$\left( a \right){\text{ 83}}{\text{.5}}$$\left( b \right){\text{ 73}}{\text{.5}}$$\left( c \right){\text{ 63}}{\text{.5}}$$\left( d \right){\text{ 53}}{\text{.5}}$

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Hint: For solving this question, we will first see the first $20$ multiples of $7$. Now we will calculate the sum of the first $n$ numbers by using the formula $\dfrac{{n\left( {n + 1} \right)}}{2}$ and then by using the average formula which will be given by $Average = \dfrac{{Sum{\text{ of all the numbers}}}}{{Total{\text{ number of terms}}}}$ . And by using this formula we will substitute it and we will get the value.

Formula used:
The average formula is given by,
$Average = \dfrac{{Sum{\text{ of all the numbers}}}}{{Total{\text{ number of terms}}}}$
Sum of first $n$ numbers,
$\dfrac{{n\left( {n + 1} \right)}}{2}$
Here, $n$ will be the number of terms.

First of all we will find the first $20$ multiples of $7$ . So the multiples will be,
$7 \times 1,7 \times 2,7 \times 3,.......,7 \times 20$ .
So the average will be calculated as
$\Rightarrow Average = \dfrac{{Sum{\text{ of all the numbers}}}}{{Total{\text{ number of terms}}}}$
Taking the term $7$ common from the numerator, we get
$\Rightarrow \dfrac{{7\left( {1 + 2 + 3 + 4 + ..... + 20} \right)}}{{20}}$
And as we know the formula for calculating the sum of $n$ numbers by using the formula $\dfrac{{n\left( {n + 1} \right)}}{2}$ .
So on substituting the values, we get
$\Rightarrow \dfrac{{7\left[ {20\left( {20 + 1} \right)} \right]}}{{2 \times 20}}$
And on solving the brace, we get the equation as
$\Rightarrow \dfrac{{7 \times 20 \times 21}}{{2 \times 20}}$
And on solving the numerator and the denominator, we get the equation as
$\Rightarrow \dfrac{{147}}{2}$
And after dividing it, we get
$\Rightarrow 73.5$
Therefore, the average of first $20$ multiples of $7$ is $73.5$ .
Hence, the option $\left( b \right)$ is correct.

Note: This question can also be calculated by using another method. In this, we will use only one formula, and the formula is given by $\dfrac{n}{2}\left( {2a + \left( {n - 1} \right)d} \right)$ . This is the formula for the sum of $n$ terms. We have the $a$ which will be the first term and in this question it is $7$ and the common difference will be given by $d$ which will be equal to $7$. So by substituting the values in the formula we will get the answer the same as we had got. So in this way also we can solve it.