Coefficient of x^{49} in expansion of (2x + 1) (2x + 3) (2x + 5)----- (2x + 99)

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7.Binomial Theorem

normal

Find the coefficient of $x^{49}$ in the expansion of $(2x + 1) (2x + 3) (2x + 5)----- (2x + 99)$

${2^{50}} \times 2500$

${2^{49}} \times 2500$

${-2^{50}} \times 2500$

${-2^{49}} \times 2500$

Solution

$(2 n+1)(2 n+3)(2 n+5) \dots(2 x+99)$

$\Rightarrow 2^{50}\left[\left(\mathrm{x}+\frac{1}{2}\right)\left(\mathrm{x}+\frac{3}{2}\right)\left(\mathrm{x}+\frac{5}{2}\right) \ldots\left(\mathrm{x}+\frac{99}{2}\right)\right]$

Coefficient of $\mathrm{x}^{49}$ is equal to $-(\mathrm{sum} \text { of roots })$

$=2^{50}\left[\frac{1}{2}+\frac{3}{2}+\frac{5}{2}+\ldots .+\frac{99}{2}\right]=2^{49} \times 2500$

Standard 11

Mathematics

$\frac{{{C_1}}}{{{C_0}}} + 2\frac{{{C_2}}}{{{C_1}}} + 3\frac{{{C_3}}}{{{C_2}}} + …. + 15\frac{{{C_{15}}}}{{{C_{14}}}} = $

medium

(IIT-1962)

If ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + … + {C_n}{x^n}$, then the value of ${C_0} + {C_2} + {C_4} + {C_6} + …..$ is

easy

The sum to $(n + 1)$ terms of the series $\frac{{{C_0}}}{2} – \frac{{{C_1}}}{3} + \frac{{{C_2}}}{4} – \frac{{{C_3}}}{5} + …$ is

normal

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)

Let ${\left( {1 + x + {x^2}} \right)^{20}}\left( {2x + 1} \right) = {a_0} + {a_1}{x^1} + {a_2}{x^2} + … + {a_{41}}{x^{41}}$ , then $\frac{{{a_0}}}{1} + \frac{{{a_1}}}{2} + …. + \frac{{{a_{41}}}}{{42}}$ is equal to

normal

Find the coefficient of $x^{49}$ in the expansion of $(2x + 1) (2x + 3) (2x + 5)----- (2x + 99)$

Solution

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If ${(1 + x)^n} = {C_0} + {C_1}x + {C_2}{x^2} + … + {C_n}{x^n}$, then the value of ${C_0} + {C_2} + {C_4} + {C_6} + …..$ is

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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 ……. .