   Question Answers

# In the usual notations prove that${{1}^{2}}.{{C}_{1}}+{{2}^{2}}.{{C}_{2}}+{{3}^{2}}.{{C}_{3}}+.........{{n}^{2}}.{{C}_{n}}=n{{\left( 2 \right)}^{n-2}}\left( n+1 \right)$.  Hint: Notice the pattern and use the binomial expansion of the expression ${{\left(1+x \right)}^{n}}$ to reach the required result. Differentiation will be needed to reach the desired answer.

Before starting the actual solution, let us try to find the general term for the above series.
So, if we observe it carefully, we will see that the series can be written as ${{C}_{1}}+{{2}^{2}}.{{C}_{2}}+{{3}^{2}}.{{C}_{3}}+.........{{n}^{2}}.{{C}_{n}}=\sum\limits_{r=1}^{n}{{{r}^{2}}{{C}_{r}}}$
Therefore, the general term is ${{T}_{r}}={{r}^{2}}.{{C}_{r}}$
Now let us start with the actual solution to the given equation.
We know that the expansion of ${{\left( 1+x \right)}^{n}}$ is:
${{\left( 1+x \right)}^{n}}=1+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+................{{C}_{n}}{{x}^{n}}$
We know that the derivative of ${{x}^{n}}$ with respect to x is $n{{x}^{n-1}}$ and also the derivative of ${{\left( 1+x \right)}^{n}}$ is $n{{\left( 1+x \right)}^{n-1}}$ . Now we will differentiate both sides of the equation. On doing so, we get
$n{{\left( 1+x \right)}^{n-1}}=0+1.{{C}_{1}}+2{{C}_{2}}{{x}^{1}}+................n{{C}_{n}}{{x}^{n-1}}$
Now multiplying the above equation by x and again differentiating both sides of the equation by x. On doing so, we get
$nx{{\left( 1+x \right)}^{n-1}}=0+1.{{C}_{1}}x+2{{C}_{2}}{{x}^{2}}+................n{{C}_{n}}{{x}^{n}}$
$\Rightarrow n{{\left( 1+x \right)}^{n-1}}+n\left( n-1 \right)x{{\left( 1+x \right)}^{n-2}}=0+1.{{C}_{1}}+{{2}^{2}}.{{C}_{2}}x+................{{n}^{2}}.{{C}_{n}}{{x}^{n-1}}$
Now we will put the value of x=1 in the above equation to prove the asked equation.
$n{{\left( 2 \right)}^{n-1}}+n\left( n-1 \right){{\left( 2 \right)}^{n-2}}=0+1.{{C}_{1}}+{{2}^{2}}.{{C}_{2}}+................{{n}^{2}}.{{C}_{n}}$

$\Rightarrow n{{\left( 2 \right)}^{n-3}}\left( 4+2n-2 \right)=0+1.{{C}_{1}}+{{2}^{2}}.{{C}_{2}}+................{{n}^{2}}.{{C}_{n}}$
$\Rightarrow n{{\left( 2 \right)}^{n-2}}\left( n+1 \right)=0+{{1}^{2}}.{{C}_{1}}+{{2}^{2}}.{{C}_{2}}+................{{n}^{2}}.{{C}_{n}}$
Hence, we have proved the equation given in the question.

Note: Be careful about the signs and try to keep the equations as neat as possible by removing the removable terms. Moreover, make sure that we learn the formulas related to different binomial expansions as in case of such questions, they are quite useful. It is prescribed that whenever we see a series including terms of the form $^{n}{{C}_{r}}$ , then always give a thought of using the binomial expansion as this would give you the answer in the shortest possible manner.
View Notes
The Difference Between an Animal that is A Regulator and One that is A Conformer  CBSE Class 11 Maths Formulas  Group Theory in Mathematics  CBSE Class 6 Maths Chapter 11 - Algebra Formulas  CBSE Class 11 Maths Chapter 16 - Probability Formulas  CBSE Class 8 Maths Chapter 11 - Mensuration Formulas  CBSE Class 12 Maths Chapter-11 Three Dimensional Geometry Formula  NCERT Book for Class 11 Maths PDF in Hindi  Coefficient of Determination  CBSE Class 11 Physics Waves Formulas  CBSE Class 12 Maths Question Paper 2020  CBSE Class 10 Maths Question Paper 2020  Maths Question Paper for CBSE Class 10 - 2011  Maths Question Paper for CBSE Class 10 - 2008  CBSE Class 10 Maths Question Paper 2017  Maths Question Paper for CBSE Class 10 - 2012  Maths Question Paper for CBSE Class 10 - 2009  Maths Question Paper for CBSE Class 10 - 2010  Maths Question Paper for CBSE Class 10 - 2007  Maths Question Paper for CBSE Class 12 - 2013  NCERT Solutions for Class 11 Maths Chapter 8 Binomial Theorem in Hindi  NCERT Solutions for Class 11 Maths Chapter 9 Sequences and Series in Hindi  NCERT Solutions for Class 11 Maths  NCERT Solutions for Class 10 English Footprints Without Feet Chapter 10 - The Book that Saved the Earth  NCERT Solutions for Class 11 Chemistry Chapter 11 The p-Block Elements In Hindi  NCERT Solutions for Class 11 Maths Chapter 11 Conic Sections in Hindi  NCERT Solutions for Class 11 Maths In Hindi  RD Sharma Class 11 Maths Solutions Chapter 23 - The Straight Lines  RD Sharma Class 11 Maths Solutions Chapter 24 - The Circle  RD Sharma Class 11 Maths Solutions Chapter 18 - Binomial Theorem  