Binomial Theorem · Mathematics · VITEEE

If $x$ is so small that $x^3$ and higher powers of $x$ may be neglected, then $\frac{(1+x)^{3 / 2}-\left(1+\frac{1}{2} x\right)^3}{(1-x)^{1 / 2}}$ may be approximate as

If $a$ and $b$ are two complex numbers, then the sum of $(n+1)$ terms of the series $a c_0-(a+d) c_1+(a+2 d) c_2-(a+3 d) c_3+$ $\_\_\_\_$ is

The value of

$$ 99^{50}-90 \cdot 98^{50}+\frac{99 \cdot 98}{1 \cdot 2}(97)^{50}-\ldots \ldots \ldots \ldots .+99 $$

is

If the coefficient of $x^2$ and $x^3$ in the expansion of $\left(1+8 x+b x^2\right)(1-3 x)^9$ in the power of $x$ are equal, then $b$ is

If the coefficients of $x^7$ and $x^8$ in the expansion of $\left[2+\frac{x}{3}\right]^n$ are equal, then value of $n$ is

Coefficient of $x^3$ in the expansion of $\left(x^2-x+1\right)^{10}\left(x^2+1\right)^{15}$ is equal to

Last three digits in $$(9)^{50}$$ be

If $$x^n=a_0+a_1(1+x)+a_2(1+x)^2+\ldots \ldots \ldots+ a_n(1+x)^n=b_0+b_1(1-x)+b_2(1-x)^2+\ldots . .+ b_n(1-x)^n$$, then for $$n=201,\left(a_{101}, b_{101}\right)$$ is equal to

If the 2nd, 3rd and 4th terms in the expansion of $$(a+b)^n$$ be $$240,720$$ and 1080 respectively, then the value of $$(n, b, a)$$ is

The coefficient of the term independent of $$x$$ in the expansion $$\left(\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right)^{10}$$ is

Which of the following is the correct principle of Mathematical induction?