Binomial Theorem for Any Index

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

A particle hanging from a spring stretches it by $1 \mathrm{cm},$ at earth's surface. How much will thw same particle stretch the spring at above the surface? The radius of the earth $\mathrm{R}=6400 \mathrm{km}.$

Explain the intensity of the gravitational field?

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______.

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

If $\left|x\right|$ is so small so that $x^2$ and higher powers of x may be neglected, then an approximate value of $\frac{{\left(1+\frac{2}{3}x\right)}^{-3}{\left(1-15x\right)}^{\frac{-1}{5}} }{{\left(2-3x\right)}^4}$ is

If $x=1+\frac{3}{1!}\times \frac{1}{6}+\frac{3\times 7}{2!}{\left(\frac{1}{6}\right)}^2+\frac{3\times 7\times 11}{3!}{\left(\frac{1}{6}\right)}^3 +\dots \infty$, then $x^4$ is

If $x=\frac{1}{5}+\frac{1.3}{5.10}+\frac{\mathrm{1.3.5}}{\mathrm{5.10.15}}+\dots \infty ,$ then $3x^2+6x=$

If $\left|x\right|<1$ then the coefficient of $x^5$ in the expansion of $\frac{3x}{\left(x-2\right) \left(x+1\right)}$ is

If $\left|x\right|<1$ then the coefficient of $x^5$ in the expansion of $\frac{3x}{\left(x-2\right) \left(x+1\right)}$ is

If ${\left(\alpha +bx\right)}^{-3}=\frac{1}{27}+\frac{1}{3}x+\dots .,$ then the ordered pair (a, b) equals to

The value of $\frac{2\frac{1}{2} }{1!}+\frac{3\frac{1}{2}}{2!}+\frac{4\frac{1}{2}}{3!}+\frac{5\frac{1}{2}}{4!}+\dots \infty$ is