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The Irrational Units Digit (Posted on 2007-05-21) Difficulty: 4 of 5
Let n be a positive integer. Find a formula for the units digit of (11+sqrt(111))^n.

See The Solution Submitted by Brian Smith    
Rating: 3.0000 (1 votes)

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Solution Solution Comment 4 of 4 |

Let A(n) = (11 + v(111))^n, and:
B(n) = (11 - v(111))^n

Then;
A(n) + B(n)
= 2( 11^n + comb(n, 2)*11^(n-2)*111 + .......)

But, we know that:

100< 111< 121
Or, 10< v(111) < 11
or, 0 < 11 - v(111)< 1
Or, 0< B(n) < 1 ...........(*)

Thus,
A(n) = 2( 11^n + comb(n, 2)*11^(n-2)*111 + .......)- B(n)
Or, D(n) > A(n) > D(n) - 1, where:
D(n) = 2( 11^n + comb(n, 2)*11^(n-2)*111 + .......) --------(**)

This gives, D(n) Mod 10
= 2(comb(n, 0) + comb(n, 2) + ........)
= 2^n

Accordingly, from (**), we obtain:
A(n) Mod 10 = 2^n -1

Thus, denoting T(n) as the units digit of n, we observe that:
T(n) Mod 10 = 2^n - 1 

Edited on May 23, 2007, 11:23 am
  Posted by K Sengupta on 2007-05-23 11:22:37

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