Binomial Expansion and Approximations

Let $a_n={\left(2+\sqrt{3}\right)}^n$. Find $\underset{n\to \infty }{\lim }\left(a_n-\left[a_n\right]\right)$, where $\left[·\right]$ denotes GIF.

Which is smaller ${\left(1.2\right)}^{4000}$ or $800$?

 Compute${\left(98\right)}^5$

Calculate ${(0.999)}^5$

Using binomial theorem, find the approximate value of $(101{)}^4$

Using binomial theorem, find the approximate value of $(1.2{)}^4$

The binomial theorem for integer exponents can be generalized to_____ exponents.

Can binomial be fractions?
 

What is binomial theorem?
 

What is binomial approximation?

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 ${\left(1+x\right)}^n=C_0+C_{1}x+C_2{x}^2+\dots +C_r{x}^r+\dots +C_n{x}^n$, then show that

$3C_1+7.C_2+11.C_3+\dots +(4n-1)C_n=1+(2n-1)2^n$.

If ${\left(1+x\right)}^n=C_0+C_{1}x+C_2{x}^2+\dots +C_r{x}^r+\dots +C_n{x}^n$, then show that

$p-{}^{n}C_1(p+q)+{}^{n}C_2(p+2q)-{}^{n}C_3(p+3q)+...$ to $n$ terms $=0$ 

If ${\left(1+x\right)}^n=C_0+C_{1}x+C_2{x}^2+\dots +C_r{x}^r+\dots +C_n{x}^n$, then show that

$a{C}_0-(a-1)C_1+(a-2)C_2-\dots +(-1{)}^n(a-n)C_n=0$

If ${\left(1+x\right)}^n=C_0+C_{1}x+C_2{x}^2+\dots +C_r{x}^r+\dots +C_n{x}^n$, then show that

$C_0+4C_1+8C_2+12C_3+\dots +4n{C}_n=1+n.2^{n+1}$

If ${\left(1+x\right)}^n=C_0+C_{1}x+C_2{x}^2+\dots +C_r{x}^r+\dots +C_n{x}^n$, then show that

$C_1-2C_2+3C_3-\dots +{\left(-1\right)}^{n-1}{C}_n=0$