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If coefficient of \( x^7 \) and \( x^8 \) are equal in expansion of \( \left( 2+\dfrac{x}{3}\right)^n \), then \( n = \)
Coefficient of
The constant term in the expansion of \(\left( \dfrac{3x^2}{2}\dfrac{1}{3x} \right)^9, x \neq0 \) is_______.
For Constant term
Coefficients of middle terms in expansion of \( \left( 2\dfrac{x^3}{3} \right)^7 \) are_______
As is odd there are terms.
& are middle terms.
&
Coefficients of & are
Middle term in expansion of \(\left( \dfrac{2}{x} 3xy\right)^{12} \) is______
is middle term.
Find the coefficient of \( t^{24} \) in the expansion \( \{(1+t^2)^{12}(1+t^{12})(1+t^{24})\} \)
Hence, coefficient of in
Coefficient of in
Coefficient of in
Coefficient of is
Coefficient of is
If \( 21^{st} \) & \( 22^{nd} \) terms in the expansion \((1+x)^{44} \) are equal, the \( x \) is equal to________
In the expansion of , & terms are equal.
i.e.,
but we know
i.e.,
Coefficients of \( 3^{rd} \) term and ________ th term are equal in the expansion \((x+y)^{13} \)
Hence,
Total terms
Coefficient of term term from last.
term
Index number of middle term in expansion of \( \left( x\frac{x^3}{5} \right)^n= \)_______., where \(n \) is even.
In the expansion , where is even, we have number of terms .
the middle term is .
Index numbr of middle term is
Index number of middle term in expansion of \( \left( 1+a+\dfrac{a^2}{u} \right)^n \) is__________
Hence is even.
The middle term is
Index number of the middle term is .
In the expansion of \(\left( a^{\frac{2}{5}} + b^{\frac{1}{3}}\right)^{35}, a \neq b \) the number of terms in which the power of \( a \) & \( b \) are integers are___________.
will be multiple of & both.
will be multiple of
& terms have integers power of & .
\(6^{th} \) term in the expansion of \(\left( \dfrac{1}{x^{\frac{8}{3}}} +x^2 \log _{10} x\right)^8 \) is \( 5600 \), then \( x = \)________.
we will find
If \( P \) & \( Q \) are coefficients of \( x^n \) in expansion of \( (1+x)^{2n} \) & \( (1+x) ^{2n1} \) then _________________
The constant term in expansion of \( \left( \dfrac{x+1}{x^{\frac{2}{3}}x^{\frac{1}{3}}+1}\dfrac{x1}{xx^{\frac{1}{2}}} \right)^{10} \) is_________
=
Now
If \( w \neq 1 \)is cube root of \( 1 \) then \( \sum_{r=0}^{100} \) \( ^{100}C_r(2+w^2)^{100r}w^r= \)_________.
The coefficient of \( x^3 \) in \( (1x+x^2)^5 \) is_______
What is the middle term in the expansion of \( \left( \dfrac{x}{2}+6y \right)^8 \)?
Hence
middle term
What is the \(3^{rd} \) term in the expansion \((x2y)^8 \)?
Hence
What is the middle term in the expansion of \( \left( x\dfrac{1}{x} \right)^{12} \)?
Hence, is the middle term in the expansion of
The coefficient of \( y \) in the expansion of \( \left( y^2+\dfrac{c}{y} \right)^5 \) is________
Now
Coefficient of
\( (1\cdot1)^{1000} \)is_____1000.
The \( 4^{th} \) term in the expansion \( (x2y)^{12} \) is__________.
If \( n \) is a true integer, then \( (\sqrt{3}+1)^{2n+1}+(\sqrt{3}1)^{2n+1} \) is__________
Since si a positive integer, assume
If \( 3^ {rd} \) term in the binomial expansion of \( (1+x)^m \) is \(\left( \dfrac{1}{8} \right)x^2 \) then the rational value of \( m \) is_______.
The greatest coefficient in the expansion of \( (1+x)^{10} \) is_________.
The coefficient of in the expansion of is and is maximum for .
Hence, the greatest coefficient
The coefficient of \( x^n \) in the expansion of \( (12x+3x^24x^3+…)^{n} \) is________.
We have,
So, the coefficient of is .
The value of \( n \) in the expansion of \( (a+b)^n \) if the first three terms of the expansion are \( 729,7290 \) & \( 30375 \), respectively is_______.
If \( \alpha \) & \(\beta \) are the roots of the equation \( x^2x+1=0 \) then the value of \( \alpha ^{2009} + \beta ^{2009} \) is________
Given
We know
So
The general term of the expansion of \( (a+b)^n \) is________
The general term of the expansion is
The coefficient of \( x^n \) in the expansion \( (1+x+x^2+โฆ)^{n} \) is_________.
Now, the coefficient of
If \( n \) is a positive integer, then \( (\sqrt{5}+1)^{2n}+1 (\sqrt{5}1)^{2n}+1 \) is________
Since is a positve integer,
In the expansion of \((a+b)^n \) , if \( n \) is even then the middle term is _________
In the expansion of , is even then is the middle term.
In the expansion of \((a+b)^n \), if \( n \) is odd then the number of middle term is /are __________
In the expansion of . If is odd then there are two middle terms i.e., & .
The number of ordere triplets of positive integers which are solution of the equation \( x+y+z=100 \) is ________.
Given, , where .
Let where .
Now,
So, the total no. of solution
In the binomial expansion of \((a+b)^n \), the coefficient of \( 4^{th} \) and \( 13 ^{th} \) terms are equal to each other, then the value of \( n \) is __________
Given the coefficient and terms are equal i.e.,
This is possible when .
Find the coefficient of \( x^{10} \) in the expansion of \( \left( \dfrac{a}{x}+bx \right)^{12} \) is_________
Given expansion is
We have to find coefficient of
Coefficient of is
Find the product of the coefficient of the middle terms in the expansion \( (x+ \dfrac {1}{x})^{11} \) is___________
Given expansion is .
Number of middle terms is .
i.e.,
Coefficient of
Coefficient of
Productof the coefficients is
Find the coefficient of \( x^n \) in the expansion \((1+x)^n \cdot (x+1)^n \).
Find the number of dissimilar terms in the expansion \(\{(a+b)+c\}^{2n+1}\{(a+b)c\}^{2n+1} \).
Given expansion
Find the coefficient of the function which is independent of \( x \) in the expansion \( \left[ \dfrac{(x^\frac{1}{3})^3+1^3}{x^\frac{2}{3}x^\frac{1}{3}+1}\dfrac{x1}{x^\frac{1}{2}(x^\frac{1}{2}1)} \right]^{10} \).
, which is independent of .
If
\( (1x)^m(1+x)^n=1+a_1x+a_2x^2+โฆ;m,n \in N, a_1=a_2=10 \) then \((m,n)= \)_________.
The expansion of \((x+\sqrt{x^31})^5+(x\sqrt{x^31})^5 \) is a polynomial of degree_______
is polynomial of degree .
(Since is the polynomial of degree )
The coefficient of \( x^r \) in the expansion of \( S= (x+3)^{n1}+(x+3)^{n2}(x+2)+(x+3)^{n3}(x+2)^2+…+(x+3)^1(x+2)^{n2}+(x+2)^{n1} \) is______
Hence
The coeffcient of the middle term in expansion of \(\left( 2x\dfrac{3}{x} \right)^4 \) is________
Hence ,
Constant term in expansion of \( \left( 1+\dfrac{x}{2}\dfrac{2}{x} \right)^4 \) is_______
The coefficient of the middle term in expansion of \( (x^310y^5)^{10} \) is
Hence middle term is
If sum of even terms are denoted by \( E \) and sum of odd term are denoted by \( O \) in expansion of \((x+a)^n \) then \( o^2E^2= \)_______.