Question

If the sum of first n natural numbers is $\dfrac{1}{5}$ times the sum of their squares, then the value of n is
$A)5$
 $B)6$
 $C)7$
 $D)8$

Answer 1

Hint: First, to solve this kind of question, we will need to know about the basic concept of sequence and series formulas. As we know the formula of the sum of first n natural numbers and their square, we will directly use this as per the given question.
That is, the sum of the first n natural numbers is $\dfrac{1}{5}$ times the sum of their squares and we need the unknown n.

Formula used:
Sum of the first n natural numbers $ = \dfrac{{n(n + 1)}}{2}$
Sum of the square n natural numbers $ = \dfrac{{n(n + 1)(2n + 1)}}{6}$
Complete step-by-step solution:
Since we know that natural numbers are the basic set in the set theory which contains the least value of zero and at the most value of undefined (infinity).
Also, we know the formulas of the Sum of the first n natural numbers $ = \dfrac{{n(n + 1)}}{2}$ , and also, we know that Sum of the square n natural numbers $ = \dfrac{{n(n + 1)(2n + 1)}}{6}$
From the given question they were given that the sum of the first n natural numbers is $\dfrac{1}{5}$ times the sum of their squares. This means we need to multiply the $\dfrac{1}{5}$ into the term sum of the square that will equal the sum of natural numbers.
Hence, we get $\dfrac{{n(n + 1)}}{2} = \dfrac{1}{5} \times \dfrac{{n(n + 1)(2n + 1)}}{6}$
Now take the common terms and then cancel each other, we have \[\dfrac{{n(n + 1)}}{2} = \dfrac{1}{5} \times \dfrac{{n(n + 1)(2n + 1)}}{6} \Rightarrow \dfrac{{n(n + 1)}}{2} = \dfrac{{n(n + 1)}}{2} \times \dfrac{1}{5} \times \dfrac{{(2n + 1)}}{3}\]
\[ \Rightarrow \dfrac{{n(n + 1)}}{2} = \dfrac{{n(n + 1)}}{2} \times \dfrac{1}{5} \times \dfrac{{(2n + 1)}}{3} \Rightarrow 1 = \dfrac{1}{5} \times \dfrac{{(2n + 1)}}{3}\]
Now by the cross multiplication, we have \[1 = \dfrac{1}{5} \times \dfrac{{(2n + 1)}}{3} \Rightarrow 15 = 2n + 1\]
By the subtraction operation, we get \[15 = 2n + 1 \Rightarrow 14 = 2n\]
By the division operation, we get \[14 = 2n \Rightarrow n = 7\]
Hence the value of the n is $7$
Therefore, the option $C)7$ is correct.

Note: The Sum of squares always 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 of n-terms, first n even terms or odd terms, set of natural numbers or a consecutive number and etc.,