How much is this sum?

tsuwal

the answer is 3^2048. How do I get there?

AlephZero

Since you are given the answer, use that information!

You have to prove that
##(2+3)(2^2+3^2)\cdots(2^{2048} + 3^{2048}) + 2^{4096} - 3^{4096} = 0##
Now, think what you can do with ##2^{4096} - 3^{4096}## ...

tsuwal

i don't know, what can i do :S?

tsuwal

and the answer is not given, it's multiple choice

mathman

a² - b² = (a-b)(a+b)

Start with a = 2²⁰⁴⁸ and b = 3²⁰⁴⁸
next repeat with a = 2¹⁰²⁴ and b = 3¹⁰²⁴
etc.
At the end you will have (2-3)(2+3). Just be careful with the sign.

tsuwal

but i got a plus sign not a minus sign...

HallsofIvy

No, its a minus sign:
##(2+3)(2^2+3^2)\cdots(2^{2048} + 3^{2048}) + 2^{4096} - 3^{4096} = 0##
AlephZero was referring to the last pair on the left.

tsuwal

now i get it. thanks!