+349 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.
