Answer

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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.

**Complete step-by-step answer:**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.

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