Interesting GIFs on mathematics…

Gone through a page that was showing some interesting GIFs as proofs… Here is the page…

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2^n = (1+2+4+…2^(n-1))+1

This is one thing that I noticed. Then tried proving it, the below is the proof.

There is a proof that we can get it through binary(To be frank, I got this proof from my friend;), the way that I tried is simple, I just added the 1’s and kept taking out the 2’s).
Take the RHS and express it in binary,
The binary of


1 is 1,
2 is 10
4 is 100

And 2^n-1 is 100..(n-1)0’s
So when I express the sum 1+2+4+…2^(n-1) as binary,
I’ll get 1111…(n-1 1’s)

Now consider the RHS,
1111….(n-1 1’s)+1

So adding 1 at the last will keeps giving a 1 carried to its preceding position.
And at last you’ll get 100..(n 0’s)

Which when expressed in decimal is 2^n
That’s it..:-)

X*(X+1)*100+25 – a perfect square

Once gone through a shortcut to find square of a number that ends with 5.

For any number that is of format X5 i.e, 10X+5. The square will be X*(X+1)*100+25.

To say it with example, say a number 45,which is 4*10+5. Hence here the X is 4. So the square will be

=4*(4+1)*100+25

=20*100+25

=2025

The process that we do above might seem tedious at first glance. But not. You just write X*(X+1) then 25 next to that. At the place of ones and tens you are getting 0 anyhow and adding 25 is same as appending 25 to X*(X+1).

Just tried to write the proof for the above.

So, we wanted to square the number 10X+5. So see how the equation evolves to get the another form.

 

=100x^2+100x+25

=100x(x+1)+25

= x(x+1)*100+25

So whatever we see in the format x(x+1)*100+25, is a perfect square with the square root as 10x+5.

Hence 9025 is a perfect square with root as 95.

13225 is a perfect square with root as 115.

38025 is a perfect square with root as 195.

We just have to see if the number, after stripping off the 25, is of format x*(x+1).