Answer
Verified
422.1k+ views
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.
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.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference Between Plant Cell and Animal Cell
Give 10 examples for herbs , shrubs , climbers , creepers
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE