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 = $\dfrac{{n\left( {n + 1} \right)}}{2}$
Also, the sum of square n natural numbers = $\dfrac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}$
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 = $\dfrac{1}{5} \times $( sum of square n natural numbers)
$ \Rightarrow $$\dfrac{{n\left( {n + 1} \right)}}{2}$ = $\dfrac{1}{5} \times \dfrac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}$
$ \Rightarrow $ 1 = $\dfrac{1}{5} \times \dfrac{{\left( {2n + 1} \right)}}{3}$
$ \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.