Interesting algebra involving the square root of a summation

mathbalarka

Math Team
Mar 2012
3,871
86
India, West Bengal
Re: Interesting algebra involving the square root of a summa

I see. But I don't think analogous arguments are very good way of proving such arguments, I want a rigorous proof.
 

agentredlum

Math Team
Jul 2011
3,372
234
North America, 42nd parallel
Re: Interesting algebra involving the square root of a summa

N has 4024 consecutive 1's. Since 4024 is even sqrt(N) has 2012 consecutive 3's before the decimal and 2012 consecutive 3's after the decimal as gelatine1 has indicated. Once the consecutive 3's are recognised, the next digit to the right (digit # 2013) must be 1 due to recursive examples posted above.

mathbalarka , I am honored that you used an idea of mine but I think the reason it didn't give you the right answer is because N is finite. Also, the consecutive 3's idea works only if N has an even number of digits. The infinteger 111... has even? odd? number of digits; see the problem?

:D
 

skipjack

Forum Staff
Dec 2006
21,478
2,470
That's far too easy. Without studying shorter numbers to see what happens, let's see how many more digits after the "1" that answers the original question you can now predict correctly without resorting to an excessively lengthy proof or using software to verify your answer. It's acceptable to guess that the first digit after the "1" is a "6", but a convincing argument is required for any further digits!