(√3+√2)^(2n)

A nice mathematics problem found on the University of Waterloo's CEMC (Centre for Education in Mathematics and Computing) website:

For each positive integer n, define an and bn to be the positive integers such that (√3 + √2)2n = an + bn√6 and (√3 − √2)2n = an − bn√6.

(a)  Determine the values of a2 and b2.

(b)  Prove that 2an − 1 < (√3 + √2)2n < 2an for all positive integers n.

(c)  Let dn be the units digit of the number (√3 + √2)2n when it is written in decimal form. Determine, with justification, the value of d1 + d2 + d3 +  ... + d1865 + d1866 + d1867.

The solution can be found here (Part B #3) if you want to check it out.