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1000th digit (Posted on 2005-04-26) Difficulty: 3 of 5
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

See The Solution Submitted by Pemmadu Raghu Ramaiah    
Rating: 3.3750 (8 votes)

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Solution Solution | Comment 7 of 30 |
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.

  Posted by Federico Kereki on 2005-04-26 23:29:31
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