Let n be a positive integer. Find a formula for the units digit of (11+sqrt(111))^n.

(11+sqrt(111))^n + (11-sqrt(111))^n = 2(11^n + C(n,2)*11^(n-2)*111 + C(n,4)*11^(n-4)*111^2 + ... )

Since factors of 11 and 111 do not affect the units digit, and 2(C(n,0)+C(n,2)+C(n,4)+ ...) = 2^n,  the units digit in question equals the units digit of 2^n - (11-sqrt(111))^n.

The units digit of 2^n is cyclic and follows the pattern 2,4,8,6,2,4,8,6, ... and since (11-sqrt(111))^n is always between zero and one, the units digit in question is 1,3,7,5,1,3,7,5, ...

 

Posted by Dennis on 2007-05-21 15:10:21