For all we know

Here's an attempt to proof the following claim that I saw in @CompSciFact's Tweet earlier today.

(in case you don't know already, I am on twitter, find me @SubhayanRM :) )

Claim:

Let F[n] be the nth Fibonacci number. 

Then F[0] + F[1] + ... + F[n] = F[n+2] - 1.

Proof: (By induction)

We deine F[ i ] = F[ i - 1 ] + F[ i - 2 ]

and  F[-1] = 1

F[-2] = 0

i.e. We start from 1.  i.e F  = 1,2,3,5,8, ... 

however the proof is valid even if the you want to start from F=0,1,1,2 .. , just change the base case :), which is trivial and certainly holds true.

"I leave it up to the reader to verify" ;) 

The inductive hypothesis and the proof of it will be the same. 

Base Case:

  for n=1

  we have F[1] = 1 = 2-1 = F[2]-1

    so the base case holds true

Inductive Hypothesis: For purposes of induction we assume that proposition

  P(n) =  F[0] + F[1] + ... + F[n ] = F[n+2] - 1 

  is true

  so,

  P(n+1)

  = F[0] + F[1] + ... + F[k ] + F[k+1]

  = F[k+2] - 1 + F[k + 1]

  = F[k+1] + F[k+2] - 1

  = F[k+3] - 1

  = F[(k+1) + 2] - 1

So by laws of mathematical induction, P(n+1) is true whenever P(n) is true.  \qed

I might be wrong... i saw a light coming on

there is a pretty cool mathematical anecdote I recently came across Dr. Lipton & Dr. Regen's blog (in case you don't know them, let's just say they's a pretty important names in CS Theory, to say the least)

Dr. Ken Regan pointed out certain "traps".. this is what he wrote..

Square root is not a function. It is written like a function, but isn’t. Here is an example of what you can “do” by conveniently forgetting this: , so take square roots of both sides. You get

This contradicts the definition .

now after looking at it for sometime,  I THINK he himself fell for a pretty bad trap!  (PLEASE FORGIVE ME AND CORRECT ME IF I AM WRONG ! )

His proof follows

                            (1)

but is it?

Recall the elementary definition of a square root: 

The square root of a number is number when squared gives back the originial number

OR

"A square root of x is number r such that r^2 = x"   Wolfram.com

Think about it..  why should ?

                             (2)           

shouldn't THIS be allowed only if both x and y are positive integers?

in general, it can be  

                             (3)

think about it again, the square both sides and you will get exactly x/y .. 

so what we can conclude is what Dr. Regan claimed is only necessarily true iff x and y are both positive.

however if we allow  sqrt{x/y} = -sqrt{x} / sqrt{y}

we still have balance in the universe where even if the square root is function,  i^2 = -1 still holds true.

------------------------------------------

BUT THEN AGAIN, I MYSELF MIGHT HAVE FALLEN VICTIM TO SOME FALLACY..
please let me know if you have any questions :)
find me at @SubhayanRM

----
i might be wrong.. it's nothing at all :)

We are a born .. born to be wild

It took me 21 years to figure out one simple thing..

The key to happiness is not necessarily success.. 

Oh wait, I actually know a lot of this stuff (movies are made of things like this..)

lets say it took me 21 years to

Caring less for everything else that does not interest me!

But that does not mean I don't want anything now, turns out I want it all.. i wan't my PhD, teach young minds, start up a company with my best friend, and good coffee !
(okay just for clarification, I am not an industry/entrepreneur guy, I want to be in the academia, the people who tell really boring jokes.. i am pretty sure I am half way there, specially with the jokes part! :D )

I like to think I have hit rock bottom, and now I have nothing to lose.. so why not have some fun now.. (after all the key idea here is to have fun, right?) Anyway, things can't possibly go any worse, and even if they do, who the hell cares, I would have had my share of fun, and in a few years down the lane, we all would be worm food or polluting the water bodies ..  so why bother living up to someone else's expectations when you can cast your own dream.

Letter to me

  

This is a reminder to the future me of what I really want.. in case that baboon thinks otherwise!

I dream to someday have my personal webpage 

And have a picture of a tiger, bear and/or lion in it.

 [to the people who understand this, feel free to make fun of me .. unless I make it! ]