If the sum of first $n$ natural number is one-fifth of the sum of their squares, then $n$ is,
(a) $5$
(b) $6$
(c) $7$
(d) $8$
Last updated date: 29th Mar 2023
•
Total views: 307.2k
•
Views today: 8.85k
Answer
307.2k+ views
Hint: Use the formulas of sequences and series to find the sum of $n$ natural numbers and their squares. Substitute these formulas in the equation which can be obtained by reading the question and then find the value of $n$.
Complete step-by-step answer:
Before proceeding with the question, we must know all the formulas of the sequences and series which will be required to solve this question.
We have a formula from which, the sum of first $r$ natural numbers (denoted by $\sum{r}$) is given by,
$\sum{r=\dfrac{r\left( r+1 \right)}{2}}....................\left( 1 \right)$
Also, we have a formula from which, the sum of the squares first $r$ natural number (denoted by $\sum{{{r}^{2}}}$) is given by,
$\sum{{{r}^{2}}=\dfrac{r\left( r+1 \right)\left( 2r+1 \right)}{6}................\left( 2 \right)}$
In the question, it is given that the sum of the first $n$ natural number is one-fifth of the sum of their squares.
$\Rightarrow $ sum of first $n$ natural numbers $=$ $\dfrac{1}{5}\times $ sum of squares of first $n$ natural numbers
$\Rightarrow $$\sum{n}=\dfrac{1}{5}\sum{{{n}^{2}}.................\left( 3 \right)}$
Substituting $r=n$ in formula $\left( 1 \right)$, the sum of first $n$ natural numbers is equal to,
$\sum{n=\dfrac{n\left( n+1 \right)}{2}}....................\left( 4 \right)$
Substituting $r=n$ in formula $\left( 2 \right)$, the sum of the squares of first $n$ natural numbers is equal to,
$\sum{{{n}^{2}}=\dfrac{n\left( n+1 \right)\left( 2n+1 \right)}{6}................\left( 5 \right)}$
Substituting $\sum{n}$ from equation $\left( 4 \right)$ and $\sum{{{n}^{2}}}$ from equation $\left( 5 \right)$ in equation $\left( 3 \right)$, we get,
$\dfrac{n\left( n+1 \right)}{2}=\dfrac{1}{5}\left( \dfrac{n\left( n+1 \right)\left( 2n+1 \right)}{6} \right)$
Cancelling $n\left( n+1 \right)$ on both the sides of the above equation, we get,
$\begin{align}
& \dfrac{1}{2}=\dfrac{\left( 2n+1 \right)}{30} \\
& \Rightarrow 15=2n+1 \\
& \Rightarrow 2n=14 \\
& \Rightarrow n=7 \\
\end{align}$
Hence, the answer is option (c).
Note: There is a possibility that one may make a mistake while applying the formula for sum of the first $n$ natural number. It is a very common mistake that one uses the formula as $\sum{n=\dfrac{n\left( n-1 \right)}{2}}$ instead of the formula $\sum{n=\dfrac{n\left( n+1 \right)}{2}}$. This may lead us to an incorrect answer.
Complete step-by-step answer:
Before proceeding with the question, we must know all the formulas of the sequences and series which will be required to solve this question.
We have a formula from which, the sum of first $r$ natural numbers (denoted by $\sum{r}$) is given by,
$\sum{r=\dfrac{r\left( r+1 \right)}{2}}....................\left( 1 \right)$
Also, we have a formula from which, the sum of the squares first $r$ natural number (denoted by $\sum{{{r}^{2}}}$) is given by,
$\sum{{{r}^{2}}=\dfrac{r\left( r+1 \right)\left( 2r+1 \right)}{6}................\left( 2 \right)}$
In the question, it is given that the sum of the first $n$ natural number is one-fifth of the sum of their squares.
$\Rightarrow $ sum of first $n$ natural numbers $=$ $\dfrac{1}{5}\times $ sum of squares of first $n$ natural numbers
$\Rightarrow $$\sum{n}=\dfrac{1}{5}\sum{{{n}^{2}}.................\left( 3 \right)}$
Substituting $r=n$ in formula $\left( 1 \right)$, the sum of first $n$ natural numbers is equal to,
$\sum{n=\dfrac{n\left( n+1 \right)}{2}}....................\left( 4 \right)$
Substituting $r=n$ in formula $\left( 2 \right)$, the sum of the squares of first $n$ natural numbers is equal to,
$\sum{{{n}^{2}}=\dfrac{n\left( n+1 \right)\left( 2n+1 \right)}{6}................\left( 5 \right)}$
Substituting $\sum{n}$ from equation $\left( 4 \right)$ and $\sum{{{n}^{2}}}$ from equation $\left( 5 \right)$ in equation $\left( 3 \right)$, we get,
$\dfrac{n\left( n+1 \right)}{2}=\dfrac{1}{5}\left( \dfrac{n\left( n+1 \right)\left( 2n+1 \right)}{6} \right)$
Cancelling $n\left( n+1 \right)$ on both the sides of the above equation, we get,
$\begin{align}
& \dfrac{1}{2}=\dfrac{\left( 2n+1 \right)}{30} \\
& \Rightarrow 15=2n+1 \\
& \Rightarrow 2n=14 \\
& \Rightarrow n=7 \\
\end{align}$
Hence, the answer is option (c).
Note: There is a possibility that one may make a mistake while applying the formula for sum of the first $n$ natural number. It is a very common mistake that one uses the formula as $\sum{n=\dfrac{n\left( n-1 \right)}{2}}$ instead of the formula $\sum{n=\dfrac{n\left( n+1 \right)}{2}}$. This may lead us to an incorrect answer.
Recently Updated Pages
Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE
