Math Challenge Question Answer

Last time, I posted this question:

 

$(2+1)(2^2+1)(2^4+1)...(2^{1024}+1)+1=?$

 

Now let's try to solve this:

 

Let's say:  $(2+1)(2^2+1)(2^4+1)...(2^{1024}+1)+1=a$

What's 2 - 1? 1, right?

How about 1 * 2? 2.

So (2 - 1)2 = 2.

Now: $(2-1)(2+1)(2^2+1)(2^4+1)...(2^{1024}+1)+1=a$

Look at it closely. Do you see something?

It's the sum and difference of two terms:

$(a+b)(a-b)=a^2-b^2$

So calculating: $(2^2-1)(2^2+1)(2^4+1)...(2^{1024}+1)+1=a$

There it is again!

Calculating, we get it again since $(2^2)^2=2^4$

After a bunch of calculating, we get: $2^{2048}-1+1=a$

1 - 1 = 0, therefore:  $2^{2048}=a$

There you go. You have the answer.

CONGRATULATIONS TO heureka FOR GETTING IT RIGHT!

Aaaaand that's it. Bye.