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Let m be the smallest positive integer such that the coefficient of \(x^2\) in the expansion of \({\left( {1 + x} \right)^2}{\left( {x + 1} \right)^3} + …. + {(1 + x)^{49}} + {(1 + mx)^{50}}\) is \({\left( {3x + 1} \right)^{51}}{C_3}\) for some +ve integer n. Then n is equal to
coefficient of in the expansion of
51n+1 must be perfect square
when n=5
m=16 when n=5
The coefficient of three consecutive terms of \({\left( {1 + x} \right)^{n + 5}}\) are in the ratio 5:10:14; then n=
Thus, and
For solving for r, we obtained r=4
Thus
The sum of coefficients of all odd degree terms in the expansion of \({\left( {x + \sqrt {{x^3} – 1} } \right)^5} + {\left( {x – \sqrt {{x^3} – 1} } \right)^5},(x > 1)\) is
Here a=x, b=
Sum of coefficient of odd degree terms = 2{110+5+5}=2
The coefficient of \({x^{10}}\) in the expansion of \({\left( {1 + x} \right)^2}{\left( {1 + {x^2}} \right)^3}{(1 + {x^3})^4}\) is
So, coefficient of
The coefficient of \({x^{2}}\) in the expansion of the product \(\left( {2 – {x^2}} \right)\left[\left( {1 + 2x + 3{x^2}} \right)^6 + \left( {1 – 4{x^2}} \right)^6\right]\) is
So, coefficient of coefficient in
– constant term in
The value of \(\left( {{}^{21}{C_1} – {}^{10}{C_1}} \right) + \left( {{}^{21}{C_2} – {}^{10}{C_2}} \right) + \left( {{}^{21}{C_3} – {}^{10}{C_3}} \right) + \left( {{}^{21}{C_4} – {}^{10}{C_4}} \right) + \cdots + \left( {{}^{21}{C_{10}} – {}^{10}{C_{10}}} \right) \) is
The coefficient of \({{x^{ – 5}}}\) in the expansion of \({\left( {{x^2} – \frac{5}{{{x^3}}}} \right)^{10}}\) is
205r= 5
5r=25
r=5
coefficient of is
If no. of terms in the expansion of \({(1 – \dfrac{2}{x} + \dfrac{4}{{{x^2}}})^n},x \ne 0\) is 28, then the sum of the coefficient of all the terms in this expansion is
The no. of terms in the expansion of is
n=6
Sum of coefficients
If \({\left( {1 + x} \right)^{2016}} + x{\left( {1 + x} \right)^{2015}} + {x^2}{(1 + x)^{2014}} + …. + {x^{2016}} = \sum\limits_{i = 10}^{2016} {{a_i}{x^i},} \) then \({{a_{17}}}\) is
let
coefficient of
If \({\left( {a + bx} \right)^{ – 3}} = \dfrac{1}{{27}} + \dfrac{1}{3}x + …,\) the ordered pair (a,b) equals to
We have,
(a,b)=(3,9)
If \({r^\text{th}}\) and \({(r + 1)^{th}}\) term in the expansion of \({\left( {p + q} \right)^n}\) are equal, then the value of \(\dfrac{{(n + 1)q}}{{r(p + q)}}\) is
If \({x^{2x}}\) occurs \({\left( {x + \frac{2}{{{x^2}}}} \right)^n}\), then n2r must be of the form.
n2r=3r
If two consecutive terms in the expansion of \({\left( {3 + 2x} \right)^{74}}\), whose coefficients are equal, are
have equal coefficients.
The coefficient of x in the expansion of \(\left( {1 + x} \right)\left( {1 + 2x} \right)\left( {1 + 3x} \right) \cdots \left( {1 + 100x} \right)\) is
The coefficient of x in the expansion of
is
=5050
The coefficient of \( x^{r}\) in the expansion of \({\left( {1 + x} \right)^{ – 2}}\) is
coefficient of is r+1.
The coefficient of \( x^{20} \) in the expansion of \({\left( {1 + 3x + 3{x^2} + {x^3}} \right)^{20}}\) is
Coefficient of is
The \({9^{{\text{th}}}}\) term of the expansion of \({\left( {3x – \frac{1}{{2x}}} \right)^8}\) is
Find the \({11^{{\text{th}}}}\) term in the expansion of \({\left( {4x – \frac{1}{{2{x^2}}}} \right)^{12}}\).
If the expansion in powers of x of the function \(\dfrac{1}{{\left( {1 – ax} \right)\left( {1 – bx} \right)}}\) is \({a_0} + {a_1}x + {a_2}{x^2} + \cdots ,\) then coefficient of \({x^n}\) is
coefficient of in the expansion is
If the \({2^{{\text{nd}}}}\) term in the expansion of \(\left( \sqrt[13]{a} + \dfrac{a}{{\sqrt {{a^{ – 1}}} }} \right)^n\) is \(14{a^{\frac{5}{2}}}\), then the value of \(\dfrac{{{}^n{C_3}}}{{{}^n{C_2}}}\) is