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$\left( a \right){\text{ 83}}{\text{.5}}$

$\left( b \right){\text{ 73}}{\text{.5}}$

$\left( c \right){\text{ 63}}{\text{.5}}$

$\left( d \right){\text{ 53}}{\text{.5}}$

Answer

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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 $73.5$ .