Binomial Theorem for Any Index

If $x$ is very small compared with unity such that $\frac{\sqrt{x+1}+\sqrt[3]{{\left(1+x\right)}^2}}{\sqrt{1+x}+\left(1+x\right)}=a+bx$ then $\left\{a,b\right\}$ is

If $x$ is small so that $x^2$ and higher powers can be neglected, then the approximate value for $\frac{{\left(1-2x\right)}^{-1}{\left(1-3x\right)}^{-2}}{{\left(1-4x\right)}^{-3}}$

If $y=3x+6x^2+10x^3+\cdots$, then $x=.........$

$1-\frac{1}{2}·\frac{1}{2}+\frac{1}{2}·\frac{3}{4}·\frac{1}{2^2}-\frac{1}{2}·\frac{3}{4}·\frac{5}{6}·\frac{1}{2^3}+\frac{1}{2}·\frac{3}{4}·\frac{5}{6}·\frac{7}{8}·\frac{1}{2^4}-......$

If the binomial expansion of $(a+bx{)}^{-2}$ is $\frac{1}{4}-3x+\dots ,$ then the values of $a$ and $b$ are

$1+\frac{1}{3}x+\frac{1\times 4}{3\times 6}{x}^2+\frac{1\times 4\times 7}{3\times 6\times 9}{x}^3+\dots$ is equal to

If ${\left(1-x\right)}^{-n}=a_0+a_{1}x+a_2{x}^2+\dots ..+a_r{x}^r+\dots$, then $a_0+a_1+a_2+\dots +a_r$ is equal to

If $-\frac{1}{3}<x<1.$ Then
$1+n\left(\frac{2x}{1+x}\right)+\frac{n(n+1)}{2!}{\left(\frac{2x}{1+x}\right)}^2+\dots \dots \dots \dots$
is equal to

For $\left|x\right|<1$, the constant term in the expansion of $\frac{1}{{\left(x-1\right)}^2\left(x-2\right)}$ is

Select the correct statement. about the variation of 'g"

Show that the value of 'g' decreases with the depth from the surface of the earth.

At what depth below the surface of the earth, the value of acceleration due to gravity becomes (i) half (ii) one-fourth of that of at the surface. The radius of the earth is $6400 \mathrm{km}$.

If $x=\frac{5}{2!3}+\frac{5·7}{3!3^2}+\frac{5·7·9}{4!3^3}+\dots ,$ then $x^2+4x=$

What is the coefficient of $\frac{y^3}{x^8}$ in $(x+y{)}^{-5},$ when $\left|\frac{y}{x}\right|<1?$

For $\left|x\right|<1,$ the coefficient of the term independent of $x$ in the expansion of $\frac{1}{{\left(x-1\right)}^2(x-2)}$is______.

Use approximation ${\left(1+x\right)}^n\approx 1+nx, \left|x\right| << 1,$to find approximate value of ${999}^{\frac{1}{3}}$

Cotyledons are also called-

The sum of the series $1+\frac{2}{3}\left(\frac{1}{8}\right)+\frac{2\times 5}{3\times 6} {\left(\frac{1}{8}\right)}^2+\frac{2\times 5\times 8}{3\times 6\times 9} {\left(\frac{1}{8}\right)}^3+\dots$

The interval in which the expansion of $\frac{3x}{\left(x-2\right) \left(x-\mathrm{1 }\right)}$is valid

The coefficient of $x^n$ in the expression of $\frac{1}{x^2-5x+6} for \left|x\right|<1$ is