If aᵣ is coefficient of x^{10-r} in Binomial expansion of (1+x)^{10} , then Σ limits

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7.Binomial Theorem

hard

If $a_r$ is the coefficient of $x^{10-r}$ in the Binomial expansion of $(1+x)^{10}$, then $\sum \limits_{r=1}^{10} r^3\left(\frac{a_r}{a_{r-1}}\right)^2$ is equal to

$4895$

$1210$

$5445$

$3025$

(JEE MAIN-2023)

Solution

$a _{ r }={ }^{10} C _{10- r }={ }^{10} C _{ r }$

$\Rightarrow \sum \limits_{ r =1}^{10} r ^3\left(\frac{{ }^{10} C _{ r }}{{ }^{10} C _{ r -1}}\right)^2=\sum \limits_{ r =1}^{10} r ^3\left(\frac{11- r }{ r }\right)^2=\sum \limits_{ r =1}^{10} r (11- r )^2$

$=\sum \limits_{ r =1}^{10}\left(121 r + r ^3-22 r ^2\right)=1210$

Standard 11

Mathematics

Coefficients of ${x^r}[0 \le r \le (n – 1)]$ in the expansion of ${(x + 3)^{n – 1}} + {(x + 3)^{n – 2}}(x + 2)$$ + {(x + 3)^{n – 3}}{(x + 2)^2} + … + {(x + 2)^{n – 1}}$

hard

If the sum of the coefficients in the expansion of $(x+y)^{n}$ is $4096,$ then the greatest coefficient in the expansion is …. .

medium

(JEE MAIN-2021)

$\mathrm{b}=1+\frac{{ }^1 \mathrm{C}_0+{ }^1 \mathrm{C}_1}{1 !}+\frac{{ }^2 \mathrm{C}_0+{ }^2 \mathrm{C}_1+{ }^2 \mathrm{C}_2}{2 !}+\frac{{ }^3 \mathrm{C}_0+{ }^3 \mathrm{C}_1+{ }^3 \mathrm{C}_2+{ }^3 \mathrm{C}_3}{3 !}+\ldots$

Let $\mathrm{a}=1+\frac{{ }^2 \mathrm{C}_2}{3 !}+\frac{{ }^3 \mathrm{C}_2}{4 !}+\frac{{ }^4 \mathrm{C}_2}{5 !}+\ldots$, Then $\frac{2 b}{a^2}$ is equal to…………………….

hard

(JEE MAIN-2024)

The value of ${\sum\limits_{r = 1}^{19} {\frac{{{}^{20}{C_{r + 1}}\left( { – 1} \right)}}{{{2^{2r + 1}}}}} ^r}$ is

normal

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)

If $a_r$ is the coefficient of $x^{10-r}$ in the Binomial expansion of $(1+x)^{10}$, then $\sum \limits_{r=1}^{10} r^3\left(\frac{a_r}{a_{r-1}}\right)^2$ is equal to

Solution

Similar Questions

Coefficients of ${x^r}[0 \le r \le (n – 1)]$ in the expansion of ${(x + 3)^{n – 1}} + {(x + 3)^{n – 2}}(x + 2)$$ + {(x + 3)^{n – 3}}{(x + 2)^2} + … + {(x + 2)^{n – 1}}$

If the sum of the coefficients in the expansion of $(x+y)^{n}$ is $4096,$ then the greatest coefficient in the expansion is …. .

The value of ${\sum\limits_{r = 1}^{19} {\frac{{{}^{20}{C_{r + 1}}\left( { – 1} \right)}}{{{2^{2r + 1}}}}} ^r}$ is

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

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