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