Question

The mean of cubes of first 15 natural numbers is ……………..
A) 980
B) 960
C) 1920
D) 1940

Answer 1

Hint:
First use the formula of sum of cube of n natural number. This formula gives you a direct sum of all first 15 natural numbers. This formula is obtained from the sum of first n natural numbers, and it is the square of sum of n natural numbers.
Mean of any given set of data is determined by dividing the sum of the given data from the number of data present in that set. It gives the average value of that set of data.

Complete step by step solution:
First we calculate the sum of the cube of the first 15 natural numbers. Sum of cube of first n natural number is determine by using the formula which is shown below,
$\sum {{n^3}} = \dfrac{{{n^2}{{\left( {n + 1} \right)}^2}}}{4}$
For first 15 natural numbers, we put $n = 15$ in above equation,
$
\sum {{n^3}} = \dfrac{{{{15}^2}{{\left( {15 + 1} \right)}^2}}}{4}\\
 = \dfrac{{{{15}^2} \times {{16}^2}}}{4}\\
 = 14400
$
Mean of cubes of the first 15 natural numbers can be calculated by dividing the sum of first 15 natural numbers from 15.
Mean$ = \dfrac{{\sum {{n^3}} }}{{15}}$
${l}
 = \dfrac{{14400}}{{15}}\\
 = 960
$

Therefore, the mean of cubes of first 15 natural numbers is 960 and option B is the correct solution for this question.

Note:
This is a direct formula based question so there is no such trick used to solve the question but the formula used in it plays a very important role to solve this question. Any mistake in the writing formula results in an incorrect solution. Lastly, don’t forget to calculate the mean by dividing the sum of cube of first 15 natural numbers from 15.