Revised text, as per Ady's comments:
If 7a + 7/a = sqrt(98)
How much is (a^777) + 1/(a^777)?
=====================
Lets divide out 7 from the first equation. Then a + 1/a = sqrt(2).
Lets define a(n) = a^n + 1/a^n
Then we know a(1)=sqrt(2), and a(2) = 0 by direct evaluation.
Now I'll introduce an identity: a(n) = a(n-1)*a(1) - a(n-2).
Then a(3) = a(2)*sqrt(2) - a(1) = -sqrt(2)
a(4) = a(3)*sqrt(2) - a(2) = -2
a(5) = a(4)*sqrt(2) - a(3) = -sqrt(2)
a(6) = a(5)*sqrt(2) - a(4) = 0
a(7) = a(6)*sqrt(2) - a(5) = sqrt(2)
a(8) = a(7)*sqrt(2) - a(6) = 2
a(9) = a(8)*sqrt(2) - a(7) = sqrt(2)
a(10) = a(9)*sqrt(2) - a(8) = 0
Okay, at this point we can see that a(1)=a(9) and a(2)=a(10), Then a(n) is periodic with period 8.
777 = 97*8+1. Then (a^777) + 1/(a^777) = a(777) = a(1) = sqrt(2).
Solution, with revised text
