Coefficient of x^{49} in expansion of (x - 1) left( {x, - ,frac{1}{2},} right) left( {x,

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7.Binomial Theorem

normal

The coefficient of $x^{49}$ in the expansion of $(x - 1)$$\left( {x\, - \,\frac{1}{2}\,} \right)$$\left( {x\, - \,\frac{1}{{{2^2}}}\,} \right)$ .....$\left( {x\, - \,\frac{1}{{{2^{49}}}}\,} \right)$ is equal to

$-2 \left( {1\, - \,\frac{1}{{{2^{50}}}}\,} \right)$

$+$ ve coefficient of $x$

$-$ ve coefficient of $x$

$-2 \left( {1\, - \,\frac{1}{{{2^{49}}}}\,} \right)$

Solution

coeff.. of $x^{49}$ in this series is

$-\left[ {1 + \frac{1}{2}\, + \frac{1}{{{2^2}}}\, + …… + \frac{1}{{{2^{49}}}}} \right]$ $=$$ – \,\left[ {\frac{{1 – \frac{1}{{{2^{50}}}}}}{{1 – \frac{1}{2}}}} \right]$ $=$$ – \,2.\,\left[ {1 – \frac{1}{{{2^{50}}}}} \right]$

Standard 11

Mathematics

If $(1 + x – 3x^2)^{2145} = a_0 + a_1x + a_2x^2 + ………$ then $a_0 – a_1 + a_2 – a_3 + ….. $ ends with

normal

$(2n + 1) (2n + 3) (2n + 5) ……. (4n – 1)$ is equal to :

normal

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

hard

(JEE MAIN-2023)

The sum of the co-efficients of all odd degree terms in the expansion of ${\left( {x + \sqrt {{x^3} – 1} } \right)^5} + {\left( {x – \sqrt {{x^3} – 1} } \right)^5},\left( {x > 1} \right)$

hard

(JEE MAIN-2018)

The sum of the coefficients in the expansion of ${(1 + x – 3{x^2})^{2163}}$ will be

easy

(IIT-1982)

The coefficient of $x^{49}$ in the expansion of $(x - 1)$$\left( {x\, - \,\frac{1}{2}\,} \right)$$\left( {x\, - \,\frac{1}{{{2^2}}}\,} \right)$ .....$\left( {x\, - \,\frac{1}{{{2^{49}}}}\,} \right)$ is equal to

Solution

Similar Questions

If $(1 + x – 3x^2)^{2145} = a_0 + a_1x + a_2x^2 + ………$ then $a_0 – a_1 + a_2 – a_3 + ….. $ ends with

$(2n + 1) (2n + 3) (2n + 5) ……. (4n – 1)$ is equal to :

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

The sum of the co-efficients of all odd degree terms in the expansion of ${\left( {x + \sqrt {{x^3} – 1} } \right)^5} + {\left( {x – \sqrt {{x^3} – 1} } \right)^5},\left( {x > 1} \right)$

The sum of the coefficients in the expansion of ${(1 + x – 3{x^2})^{2163}}$ will be

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