Rare expressions

I present the following two expressions. One gives approximately a squared number and the other approximately a triangular number.

I hope you like the expressions.

Have a nice day !!

These results are already blogged about by me.

So in the just above if we increase k by 1 ( k is large) then the two results (with k and k plus 1) when divided give n squared in the first case and 2 n squared plus n in the second.

Essentially with the binomial in the RHS do the following operation

Put k as say 1000. Do the binomial sigma.

Then put k as 1001. Do the binomial sigma.

If we divide, then we get n squared in the first case and 2 n squared plus n in the second.

Is it not interesting?

Is it true for all functions?

Yes, it seems so.

So, the following seems to be the case. 

Here is a workout on Wolfram Alpha to support my just above statement

The result by dividing above two is 3. And 3 is what the function of n all about.