If {left( {1 + x} right)^n} = {c_0} + {c₁}x + {c₂}{x^2} + {c₃}{x^3} + ...... + {cₙ}{x^

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7.Binomial Theorem

normal

If ${\left( {1 + x} \right)^n} = {c_0} + {c_1}x + {c_2}{x^2} + {c_3}{x^3} + ...... + {c_n}{x^n}$ , then the value of ${c_0} - 3{c_1} + 5{c_2} - ........ + {( - 1)^n}\,(2n + 1){c_n}$ is

$\left( {n - 1} \right){.2^n}$

$0$

$\left( {1 - 2n} \right){.2^{n - 1}}$

$\left( {1 - n} \right){.2^n}$

Solution

$(1+x)^{n}=C_{0}+C_{1} x+C_{2} x^{2}+\ldots . C_{n} x^{n}$

$\mathrm{x} \rightarrow \mathrm{x}^{2}$

$\left(1+x^{2}\right)^{n}=C_{0}+C_{1} x^{2}+C_{2} x^{4}+\ldots . C_{n} x^{2 n}$

Multiplying $x,$

$\mathrm{x}\left(1+\mathrm{x}^{2}\right)^{\mathrm{n}}=\mathrm{C}_{0} \mathrm{x}+\mathrm{C}_{1} \mathrm{x}^{3}+\mathrm{C}_{2} \mathrm{x}^{5}+\ldots . \mathrm{C}_{\mathrm{n}} \mathrm{x}^{2 \mathrm{n}+1}$

Differentiating both sides w.r.t. $\mathrm{x}$, then

$x.n.{\left( {1 + {x^2}} \right)^{n – 1}} \cdot 2x + {\left( {1 + {x^2}} \right)^n} = {C_0} + 3{C_1}{x^2} + 5{C_2}{x^4}$

$ + \ldots \ldots + (2n + 1){C_n}{x^{2n}}$

Put $x=i$ in both sides

$0+0=\mathrm{C}_{0}-3 \mathrm{C}_{1}+5 \mathrm{C}_{2}-\ldots \ldots .(-1)^{\mathrm{n}}(2 \mathrm{n}+1) \mathrm{C}_{\mathrm{n}}=0$

Standard 11

Mathematics

For $x\, \in \,R\,,\,x\, \ne \, – 1,$ if ${(1 + x)^{2016}} + x{(1 + x)^{2015}} + {x^2}{(1 + x)^{2014}} + ….{x^{2016}} = \sum\limits_{i = 0}^{2016} {{a_i\,}{\,x^i}} ,$ then $a_{17}$ is equal to

hard

(JEE MAIN-2016)

Let ${ }^{n} C_{r}$ denote the binomial coefficient of $x^{r}$ in the expansion of $(1+ x )^{ n }.$

If $\sum_{ k =0}^{10}\left(2^{2}+3 k \right){ }^{ n } C _{ k }=\alpha .3^{10}+\beta \cdot 2^{10}, \alpha, \beta \in R$ then $\alpha+\beta$ is equal to ……. .

hard

(JEE MAIN-2021)

If $\sum_{ r =0}^5 \frac{{ }^{11} C _{2 r +1}}{2 r +2}=\frac{ m }{ n }, \operatorname{gcd}( m , n )=1$, then $m – n$ is equal to _____

hard

(JEE MAIN-2025)

If ${(1 – x + {x^2})^n} = {a_0} + {a_1}x + {a_2}{x^2} + …. + {a_{2n}}{x^{2n}}$, then ${a_0} + {a_2} + {a_4} + …. + {a_{2n}} = $

hard

The coefficent of $x^7$ in the expansion of ${\left( {1 – x – {x^2} + {x^3}} \right)^6}$ is

hard

(AIEEE-2011)

If ${\left( {1 + x} \right)^n} = {c_0} + {c_1}x + {c_2}{x^2} + {c_3}{x^3} + ...... + {c_n}{x^n}$ , then the value of ${c_0} - 3{c_1} + 5{c_2} - ........ + {( - 1)^n}\,(2n + 1){c_n}$ is

Solution

Similar Questions

For $x\, \in \,R\,,\,x\, \ne \, – 1,$ if ${(1 + x)^{2016}} + x{(1 + x)^{2015}} + {x^2}{(1 + x)^{2014}} + ….{x^{2016}} = \sum\limits_{i = 0}^{2016} {{a_i\,}{\,x^i}} ,$ then $a_{17}$ is equal to

Let ${ }^{n} C_{r}$ denote the binomial coefficient of $x^{r}$ in the expansion of $(1+ x )^{ n }.$ If $\sum_{ k =0}^{10}\left(2^{2}+3 k \right){ }^{ n } C _{ k }=\alpha .3^{10}+\beta \cdot 2^{10}, \alpha, \beta \in R$ then $\alpha+\beta$ is equal to ……. .

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Let ${ }^{n} C_{r}$ denote the binomial coefficient of $x^{r}$ in the expansion of $(1+ x )^{ n }.$

If $\sum_{ k =0}^{10}\left(2^{2}+3 k \right){ }^{ n } C _{ k }=\alpha .3^{10}+\beta \cdot 2^{10}, \alpha, \beta \in R$ then $\alpha+\beta$ is equal to ……. .