   Question Answers

# Show that the middle term in the expansion of ${\left( {1 + x} \right)^n}$is $6{x^2}$ if n = 4.  Hint - There can be two methods to solve this problem, one is based upon the general formula of expansion of ${\left( {1 + x} \right)^n}$ using the binomial expansion and the other one focuses on direct formula to find the middle term in expansion of ${\left( {1 + x} \right)^n}$ depending upon whether n is even or odd.

Now let’s use the direct formula for finding the middle term in the expansion of ${\left( {1 + x} \right)^n}$.
Here n =4 (given in question)
Clearly n is even thus the middle term in expansion of ${\left( {1 + x} \right)^n}$is $^n{C_{\dfrac{n}{2}}}{x^{\dfrac{n}{2}}}$……………… (1)
So let’s directly put the value of n in equation (1) we get
Middle term will be $^4{C_{\dfrac{4}{2}}}{x^{\dfrac{4}{2}}} = {{\text{ }}^4}{C_2}{x^2}$………………….. (2)
Now The formula for $^n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}$…………………. (3)
Using equation three we get
$^4{C_2} = \dfrac{{4!}}{{2!\left( {4 - 2} \right)!}} \\ \Rightarrow \dfrac{{4!}}{{2!\left( 2 \right)!}} \\$
Now $n! = n\left( {n - 1} \right)\left( {n - 2} \right)\left( {n - 3} \right).........\left( {n - r} \right)$ such that $r < n$
Using this concept we get
$^4{C_2} = \dfrac{{4 \times 3 \times 2 \times 1}}{{2 \times 1 \times 2 \times 1}} \\ \Rightarrow 6 \\$
Hence putting it in equation (2) we get
Middle term of ${\left( {1 + x} \right)^n}$is $6{x^2}$

Note – Now let’s talk about a method in which we can use the entire expansion of ${\left( {1 + x} \right)^n}$using binomial theorem. The expansion of ${\left( {1 + x} \right)^n}$is$1 + nx + \dfrac{{n\left( {n - 1} \right)}}{2}{x^2} + \dfrac{{n\left( {n - 1} \right)\left( {n - 2} \right)}}{{3!}}{x^3} + ............\infty$. Hence after completely expanding till the given value of n, find the middle term in that series, this will also give the right answer.
View Notes
Binomial Expansion Formula  Significance of Genetics in the Process of Evolution  The Difference Between an Animal that is A Regulator and One that is A Conformer  The Idea of Time  The Perimeter of Rectangle Formula  Dark Side of the Moon  The Ghat of the Only World Summary  What are the Functions of the Human Skeletal System?  Measuring the Rate of Change of Motion  The Converse of Pythagoras Theorem  Previous Year Question Paper of CBSE Class 10 English  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  NCERT Solutions Class 11 English Woven Words Prose Chapter 4 The Adventure of the Three Garridebs  NCERT Solutions for Class 11 Biology Chapter 8 Cell: The Unit of Life in Hindi  NCERT Solutions for Class 10 English Footprints Without Feet Chapter 10 - The Book that Saved the Earth  NCERT Solutions for Class 11 English Snapshots Chapter 6 - The Ghat of the Only World  NCERT Solutions for Class 11 English Hornbill Chapter 4 - Landscape of the Soul  RD Sharma Class 11 Maths Solutions Chapter 23 - The Straight Lines  RS Aggarwal Class 11 Solutions Chapter-16 Conditional Identities Involving the Angles of a Triangle  RD Sharma Class 11 Maths Solutions Chapter 24 - The Circle  NCERT Solutions for Class 9 Science Chapter 4 Structure of The Atom in Hindi  NCERT Solutions for Class 11 Chemistry Chapter 11 The p-Block Elements In Hindi  