What is the 1000th digit to the right of the decimal point in the decimal representation of (1+√2)^3000?
This problem can be solved by algebra alone, without the need for computers or calculators
If you add (1-√2)^3000 to (1+√2)^3000, you get an integer -- this can
be seeing by applying Newton's formula. So, if we can estimate the
fractional part of (1-√2)^3000 we can subtract it from 1.00000... and
answer the question.
As (1-√2)< 0.42, (1-√2)^3000<0.42^3000; by
logarithms, log(0.42)=-0.37... so 0.42^3000< 10^(0.37x3000)<
10^1000. So, we are subtracting a number that has over a thousand
zeroes after the decimal point from an integer number; the result has
then over a thousand nines after the decimal point.