The sum of the coefficients of the first three terms in the expansion of $\left(x-\frac{3}{x^{2}}\right)^{m}, x \neq 0, m$ being a natural number, is $559 .$ Find the term of the expansion containing $x^{3}$
The sum of the coefficients of the first three terms in the expansion of $\left(x-\frac{3}{x^{2}}\right)^{m}, x \neq 0, m$ being a natural number, is $559 .$ Find the term of the expansion containing $x^{3}$
The coefficients of the first three terms of ${\left( {x - \frac{3}{{{x^2}}}} \right)^m}$ are $^m{C_0},( - 3){\,^m}{C_1}$ and $\,9{\,^m}{C_2}$. Therefore, by the given condition, we have
$^m{C_0} - 3{\,^m}{C_1} + 9{\,^m}{C_2} = 559,$ i.e., $1 - 3m + \frac{{9m(m - 1)}}{2} = 559$
which gives $m=12$ ( $m$ being a natural number).
Now ${T_{r + 1}} = {\,^{12}}{C_r}{x^{12 - r}}{\left( { - \frac{3}{{{x^2}}}} \right)^r} = {\,^{12}}{C_r}{( - 3)^r} \cdot {x^{12 - 3r}}$
Since we need the term containing $x^{3}$, so put $12-3 r=3$ i.e., $r=3$
Thus, the required term is ${\,^{12}}{C_3}{( - 3)^3}{x^3},$ i.e., $-5940 x^{3}$
Similar Questions
If the coefficient of the middle term in the expansion of ${(1 + x)^{2n + 2}}$ is $p$ and the coefficients of middle terms in the expansion of ${(1 + x)^{2n + 1}}$ are $q$ and $r$, then
If the coefficient of the middle term in the expansion of ${(1 + x)^{2n + 2}}$ is $p$ and the coefficients of middle terms in the expansion of ${(1 + x)^{2n + 1}}$ are $q$ and $r$, then
Number of rational terms in the expansion of ${\left( {\sqrt 2 \,\, + \,\,\sqrt[4]{3}} \right)^{100}}$ is :
Number of rational terms in the expansion of ${\left( {\sqrt 2 \,\, + \,\,\sqrt[4]{3}} \right)^{100}}$ is :
Find the coefficient of $a^{4}$ in the product $(1+2 a)^{4}(2-a)^{5}$ using binomial theorem.
Find the coefficient of $a^{4}$ in the product $(1+2 a)^{4}(2-a)^{5}$ using binomial theorem.
The ratio of the coefficient of the middle term in the expansion of $(1+x)^{20}$ and the sum of the coefficients of two middle terms in expansion of $(1+x)^{19}$ is $....$
The ratio of the coefficient of the middle term in the expansion of $(1+x)^{20}$ and the sum of the coefficients of two middle terms in expansion of $(1+x)^{19}$ is $....$
- [JEE MAIN 2021]
Middle term in the expansion of ${(1 + 3x + 3{x^2} + {x^3})^6}$ is
Middle term in the expansion of ${(1 + 3x + 3{x^2} + {x^3})^6}$ is