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Right and Left Digits (Posted on 2016-07-17) Difficulty: 3 of 5
What is:
(a) the first digit to the left of the decimal point
(b) the first digit to the right of the decimal point
in the base ten expansion of (√2 + √5)2000?

No Solution Yet Submitted by K Sengupta    
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Solution Solution Comment 1 of 1
This is similar to The Irrational Units Digit, so I will adapt its solution for this problem.

sqrt(2)+sqrt(5) and its conjugate sqrt(5)-sqrt(2) are roots of x^4 - 14x^2 + 9 = 0.  Then the sum of the powers of sqrt(2)+sqrt(5) and sqrt(5)-sqrt(2) obey the recursion s(2n+4) = 14*s(2n+2) - 9*s(2n).

Direct calculation finds s(2) = 7 and s(4) = 89.  The sequence continues 7, 89, 1183, 15761, 210007, ...  The last digit repeats in a cycle of 4: 7, 9, 3, 1, 7, 9, 3, 1, ...  Then the last digit of s(2000), which is the last digit of [sqrt(2)+sqrt(5)]^2000 + [sqrt(5)-sqrt(2)]^2000, is 1.

sqrt(5)-sqrt(2) is between 0 and 1, then [sqrt(5)-sqrt(2)]^2000 is also between 0 and 1.  Therefore the digits on either side of the decimal point are 0.9.

  Posted by Brian Smith on 2016-07-17 21:51:53
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