Question

If the sum of the first n natural numbers is 1/5 times the sum of their squares, then the value of n is
A. 5
B. 6
C. 7
D. 8

Answer 1

Hint: To solve this question, we need to know the basic formula related to the sequence and series. As we know the formula of Sum of first n natural numbers and their square, we will directly use this as per question statements i.e. the sum of the first n natural numbers is 1/5 times the sum of their squares and find the value of n.

Complete step-by-step answer:
We know that,
Sum of first n natural numbers = ${\displaystyle \frac{n\left(n+1\right)}{2}}$
Also, the sum of square n natural numbers = ${\displaystyle \frac{n\left(n+1\right)\left(2n+1\right)}{6}}$
According to the question,
The sum of the first n natural numbers is 1/5 times the sum of their squares.
Sum of first n natural numbers = ${\displaystyle \frac{1}{5}}\times$( sum of square n natural numbers)
$\Rightarrow$${\displaystyle \frac{n\left(n+1\right)}{2}}$ = ${\displaystyle \frac{1}{5}}\times {\displaystyle \frac{n\left(n+1\right)\left(2n+1\right)}{6}}$
$\Rightarrow$ 1 = ${\displaystyle \frac{1}{5}}\times {\displaystyle \frac{\left(2n+1\right)}{3}}$
$\Rightarrow$ 15 = $2n+1$
$\Rightarrow$ $2n=14$
$\Rightarrow$ $n=7$
Therefore, the value of n is 7.
Thus, option (c) is the correct answer.

Note: Sum of squares refers to the sum of the squares of numbers. It is basically the addition of squared numbers. The squared terms could be 2 terms, 3 terms, or ‘n’ number of terms, first n even terms or odd terms, set of natural numbers or consecutive numbers, etc.