Answer
424.8k+ views
Hint: Here we have to compare coefficient of ${x^n}$in both the given expression. For that we will get the expression for (r+1)th term of both ${(1 + x)^{2n}}$and ${(1 + x)^{2n - 1}}$. And from there we will get the coefficient of ${x^n}$.
Complete step-by-step answer:
Step-1
We know that the general form of ${(a + b)^n}$ is
${T_{r + 1}} = {}^n{C_r}{a^{n - r}}{b^r}$
Where, ${T_{r + 1}}$=(r+1)th term
n = highest power of the expression
step-2
In the expression ${(1 + x)^{2n}}$, a=1, b=x, n=2n
The general form of it is,
${T_{r + 1}} = {}^{2n}{C_r}{1^{2n - r}}{x^r}$
$ \Rightarrow {T_{r + 1}} = {}^{2n}{C_r}{x^r}$……………(1)
Step-3
For coefficient of ${x^n}$, putting r=n in equation (1) we get,
${T_{n + 1}} = {}^{2n}{C_n}{x^n}$…………….(2)
Step-4
The coefficient of ${x^n}$ is${}^{2n}{C_n}$
In the expression ${(1 + x)^{2n - 1}}$, a=1, b=x, n=2n-1
The general form of it is,
${T_{r + 1}} = {}^{2n - 1}{C_r}{1^{2n - 1 - r}}{x^r}$
$ \Rightarrow {T_{r + 1}} = {}^{2n - 1}{C_r}{x^r}$……………(3)
Step-5
For coefficient of ${x^n}$, putting r=n in equation (3) we get,
${T_{n + 1}} = {}^{2n - 1}{C_n}{x^n}$……………(4)
The coefficient of ${x^n}$ is ${}^{2n - 1}{C_n}$
Step-6
We are asked to prove that the coefficient of ${x^n}$ in the expression of ${(1 + x)^{2n}}$ is twice of the coefficient of the ${x^n}$ in the expression of ${(1 + x)^{2n - 1}}$
i.e. ${}^{2n}{C_n}$= 2${}^{2n - 1}{C_n}$
step-7
Simplifying left hand side,
${}^{2n}{C_n}$
$ = \dfrac{{2n!}}{{n!(2n - n)!}}$
$ = \dfrac{{2n!}}{{n!n!}}$
Step-8
Again simplifying right hand side we get,
2${}^{2n - 1}{C_n}$
$ = 2 \times \dfrac{{(2n - 1)!}}{{n!(2n - 1 - n)!}}$
$ = 2 \times \dfrac{{(2n - 1)!}}{{n!(n - 1)!}}$
Step-9
Multiplying and dividing by n we get,
$ = 2 \times \dfrac{{(2n - 1)!}}{{n!(n - 1)!}} \times \dfrac{n}{n}$
$ = \dfrac{{2n(2n - 1)!}}{{n!n(n - 1)!}}$
$ = \dfrac{{2n!}}{{n!n!}}$
Step-10
It is proved that L.H.S = R.H.S
Hence it is proved that the coefficient of ${x^n}$ in the expression of ${(1 + x)^{2n}}$ is twice of the coefficient of the ${x^n}$ in the expression of ${(1 + x)^{2n - 1}}$.
Note: This is a binomial sequence and series question. Be cautious while solving expansion of general form of expression and while comparing them.
Binomial sequences can be used to prove results and solve problems in combinatorics, algebra, calculus, and many other areas of mathematics.
Complete step-by-step answer:
Step-1
We know that the general form of ${(a + b)^n}$ is
${T_{r + 1}} = {}^n{C_r}{a^{n - r}}{b^r}$
Where, ${T_{r + 1}}$=(r+1)th term
n = highest power of the expression
step-2
In the expression ${(1 + x)^{2n}}$, a=1, b=x, n=2n
The general form of it is,
${T_{r + 1}} = {}^{2n}{C_r}{1^{2n - r}}{x^r}$
$ \Rightarrow {T_{r + 1}} = {}^{2n}{C_r}{x^r}$……………(1)
Step-3
For coefficient of ${x^n}$, putting r=n in equation (1) we get,
${T_{n + 1}} = {}^{2n}{C_n}{x^n}$…………….(2)
Step-4
The coefficient of ${x^n}$ is${}^{2n}{C_n}$
In the expression ${(1 + x)^{2n - 1}}$, a=1, b=x, n=2n-1
The general form of it is,
${T_{r + 1}} = {}^{2n - 1}{C_r}{1^{2n - 1 - r}}{x^r}$
$ \Rightarrow {T_{r + 1}} = {}^{2n - 1}{C_r}{x^r}$……………(3)
Step-5
For coefficient of ${x^n}$, putting r=n in equation (3) we get,
${T_{n + 1}} = {}^{2n - 1}{C_n}{x^n}$……………(4)
The coefficient of ${x^n}$ is ${}^{2n - 1}{C_n}$
Step-6
We are asked to prove that the coefficient of ${x^n}$ in the expression of ${(1 + x)^{2n}}$ is twice of the coefficient of the ${x^n}$ in the expression of ${(1 + x)^{2n - 1}}$
i.e. ${}^{2n}{C_n}$= 2${}^{2n - 1}{C_n}$
step-7
Simplifying left hand side,
${}^{2n}{C_n}$
$ = \dfrac{{2n!}}{{n!(2n - n)!}}$
$ = \dfrac{{2n!}}{{n!n!}}$
Step-8
Again simplifying right hand side we get,
2${}^{2n - 1}{C_n}$
$ = 2 \times \dfrac{{(2n - 1)!}}{{n!(2n - 1 - n)!}}$
$ = 2 \times \dfrac{{(2n - 1)!}}{{n!(n - 1)!}}$
Step-9
Multiplying and dividing by n we get,
$ = 2 \times \dfrac{{(2n - 1)!}}{{n!(n - 1)!}} \times \dfrac{n}{n}$
$ = \dfrac{{2n(2n - 1)!}}{{n!n(n - 1)!}}$
$ = \dfrac{{2n!}}{{n!n!}}$
Step-10
It is proved that L.H.S = R.H.S
Hence it is proved that the coefficient of ${x^n}$ in the expression of ${(1 + x)^{2n}}$ is twice of the coefficient of the ${x^n}$ in the expression of ${(1 + x)^{2n - 1}}$.
Note: This is a binomial sequence and series question. Be cautious while solving expansion of general form of expression and while comparing them.
Binomial sequences can be used to prove results and solve problems in combinatorics, algebra, calculus, and many other areas of mathematics.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)