Question

What is the average of the squares of the first 19 natural numbers?
(A) 124
(B) 127.5
(C) 130
(D) 133.5

Answer 1

Hint: Firsts of all, use the formula for the summation of the first n natural numbers, Summation = \[\dfrac{n\left( n+1 \right)\left( 2n+1 \right)}{6}\] , and calculate the summation of the first 19 natural numbers. To find the average of the squares of the first 19 natural numbers, use the formula, \[\dfrac{Sum\,of\,square\,of\,first\,19\,natural\,numbers}{19}\] . Now, solve it further to get the required answer.

Complete step-by-step answer:
According to the question, we are asked to find the squares of the first 19 natural numbers.
First of all, we need to know about natural numbers.
We know that natural numbers are all those numbers that are greater than or equal to 1. It should not be a decimal point number …………………………………..(1)
Also, a natural number cannot be equal to zero or a negative number …………………………………….(2)
From equation (1) and equation (2), we can say that the natural numbers are all those positive integral numbers excluding zero.
Now, we have to find the average of the first 19 natural numbers.
We know the formula for average, Average of the first 19 natural numbers = \[\dfrac{Sum\,of\,square\,of\,first\,19\,natural\,numbers}{19}\] …………………………………………….(3)
From equation (3), we can observe that we need to find the summation of the first 19 natural numbers.
We know the formula for the summation of the first n natural numbers, Summation = \[\dfrac{n\left( n+1 \right)\left( 2n+1 \right)}{6}\] …………………………………………….(4)
Since we require the summation of the square of the first 19 natural numbers so, we have put
\[n=19\] in the formula shown in equation (4).
On putting \[n=19\] in equation (4), we get
Summation of the square of the first 19 natural numbers = \[\dfrac{19\left( 19+1 \right)\left( 2\times 19+1 \right)}{6}=\dfrac{19\times 20\times 39}{6}\] = 2470 ………………………………………….(5)
Now, from equation (3) and equation (5), we get
Average of the first 19 natural numbers = \[\dfrac{2470}{19}=130\] .
Therefore, the average of the first 19 natural numbers is 130.
Hence, the correct option is (C).

So, the correct answer is “Option (C)”.

Note: For this where we require the summation of the square of the first n natural numbers, don’t approach by squaring all the numbers starting from 1 to n and then by adding them all. Approach it by using the formula for the summation of the first n natural numbers, \[\dfrac{n\left( n+1 \right)\left( 2n+1 \right)}{6}\] .