How to solve the question: If $A=(2+1)(2^2+1)(2^3+1)......(2^{2048}+1)$, what shall be the value of $(A+1)^{1/2048}$. I have not been able to think of anything in any direction, Tobe honest!
Asked
Active
Viewed 57 times
3
-
Are you sure that the original product is $(2+1)(2^2+1)(2^3+1)(2^4+1)\cdots$ , not $(2+1)(2^2+1)(2^4+1)(2^8+1)\cdots$? – Mythomorphic Sep 05 '16 at 15:59
-
Yes sir. That was kind of my initial doubt as well. But that is how the question is !!! Hope that clarifies. – asabhish Sep 05 '16 at 16:42
1 Answers
0
First, I think the third term in your expression should be $2^{2^2}+1$ as hkmather802 pointed out. The general term should be $2^{2^n}+1$.
If this is the case, here is a solution.
Write $A$ as $A=(2-1)A$ and use difference square formula $a^2-b^2=(a+b)(a-b)$, you can get $$A=2^{4096}-1.$$ Thus $$(A+1)^{1/2048}=(2^{4096})^{1/2048}=2^2=4.$$
If it is your case, I have no idea.
Q-Zhang
- 1,608