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\(\sum\limits_{r = 0}^n {{}^n{C_r}{4^r}} = \)
putting x=4
\({(1.1)^{10000}}\) is โฆ.\({2^{10}}\)
other the terms
=1+1000+100+other the terms
=1101+other the terms
\({{{10}^{th}}}\) term in expansion of \({(2{x^2} + \frac{1}{{{x^2}}})^{25}}\) isโฆ.
If coefficient of \({x^7}\) in \({(a{x^2} + \frac{a}{{bx}})^{11}}\) equals the cofficient of \({x^{ – 7}}\) in \({(a{x^2} – \frac{1}{{bx}})^{11}}\) then
If \({(r + 1)^{th}}\) term in expansion of \({(2{x^2} + \frac{5}{{{x^2}}})^{10}}\) is constant then r =
204r=0
=4r=20
r=5
If \({r^{{\text{th}}}}\) term in expansion of \({\left( {x + \frac{1}{{2x}}} \right)^{12}}\) is constant then r =
2r=14
r=7
Coefficient of \({x^{ – 3}}\) in expansion of \({\left( {x – \dfrac{a}{x}} \right)^{11}}\) is
,Here n=11
112r=3
2r=14
r=7
coefficient of
The \({{n^ \text {th}}}\) term in expansion of \({\left( {{x^3} – \frac{1}{{{x^3}}}} \right)^{12}}\) is constant term, then find r
Find the coefficient of \( x^{5}\) in expansion of \({\left( {{x^3} – \frac{2}{{{x^2}}}} \right)^7}\).
Coefficient of \( x^{11}\) in the expansion of \({\left( {1 + {x^2}} \right)^4}{\left( {1 + {x^3}} \right)^7}{\left( {1 + {x^2}} \right)^{12}}\) is
Coefficient of in the expansion of
is
+ + +
=
462+140+504+7=1113
For r=0,1,โฆ,10. Let \({A_r},{B_r}\) and \({C_r}\) denote respectively, the coeffiecent of \({x^r}\) in expansion of \({\left( {1 + x} \right)^{10}},{\left( {1 + x} \right)^{20}}\) and \({\left( {1 + x} \right)^{30}}\). Then \(\sum\limits_{r = 1}^n {{A_r}({B_{10}}{B_r} – {C_{10}}{A_r})} \) is equal to
= coeffecient of in the expansion of and = coffiecient of in
We have ,
The Expression \({\{ x + {\left( {{x^3} – 1} \right)^{1/2}}\} ^5} + {\{ x – {\left( {{x^3} – 1} \right)^{1/2}}\} ^5}\) is a polynomial of degree.
The degree of the polynomial is 7.
Since degree of is 7.
The sum \(\sum\limits_{i = 1}^m {\left( \begin{array}10\\i\end{array} \right)} \left( \begin{array}20\\m – i\end{array} \right)\), where \( \dfrac pq =0 \) of p>q is maximum when m is
is the coefficient of in the expansion of .
Coefficient of in expansion of .
is maximum when
In expansion of \({\left( {a – b} \right)^n},n \ge 5,\) the sum of the \({5^\text {th}}\) and \({6^\text {th}}\) term is zero, then \(\dfrac{a}{b}\) equals to
\( = {}^n{C_4}{(a)^{n – 4}}{b^4}\)
In the expansion of \({\left( {1 + x} \right)^m}{\left( {1 – x} \right)^n}\), the coefficient of x and \( x^2 \) are 3 and 6 respectively, then m equals to
If \({{C_r}}\) stands for \({}^n{C_r}\) then the sum of the series \({\frac{{2(\frac{n}{2})!(\frac{n}{2})!}}{{n!}}}\)\({[C_0^2 – 2C_1^2 + ….. + {{\left( { – 1} \right)}^n}\left( {n + 1} \right)C_n^2]}\) where n is an ever positive integer, is _______
n=2k,kN
\({[{}^{\left( {2k + 1} \right)}C_0^2 – 2C_1^2 + …. + C_{2k}^2]}\)
The coefficient of \( x^4 \) in \({\left( {\frac{x}{2} – \frac{3}{{{x^2}}}} \right)^{10}}\) is
cofficient of
The cofficient of \({\left( {3r} \right)^{th}}\) and \({(r + 2)^{th}}\) terms in the expansion of \({\left( {1 + x} \right)^{2n}}\) are equal, then (where r>1, n>1)
3r1=r+1 or 2n=(3r1)+(r+1)
2r=2 or 2n=4r
r=1 or n=2r
But r>1.
For a position integer n, Let \( a(n) = 1 + \frac{1}{2} + \frac{1}{3} + …… + \frac{1}{{{2^n} – 1}}\), then
Let \(x = {({}^{10}{C_1})^2} + 2{({}^{10}{C_2})^2} + …. + {}^{10}{({}^{10}{C_{10}})^2}\), where \({}^{10}{C_r}, r \in \{ 1,2,…,10\} \) denote binomial cofficients. Then, the value of \(\dfrac{1}{{1430}}x\) is ____________