Scalable Fibonacci Solutions
Name:
Anonymous
2011-10-03 18:50
Name:
Anonymous
2011-10-03 23:16
>>5
f_(2(n+1)+1) = f_(2n+3)
= f_(2n+2) + f_(2n+1)
= f_(2n+1) + f_(2n) + f_(2n+1)
= 2f_(2n+1) + f_(2n)
= 2f_(2n+1) + f_(2n+1) - f_(2n-1)
= 3f_(2n+1) - f_(2(n-1)+1)
= 3[(f_n)^2 + (f_(n+1))^2] - [(f_(n-1))^2 + (f_n)^2] Using Strong induction..
= 3(f_n)^2 + 3(f_(n+1))^2 - (f_(n-1))^2 - (f_n)^2
= 2(f_n)^2 + 3(f_(n+1))^2 - (f_(n-1))^2
= 2(f_n)^2 + 3(f_(n+1))^2 - (f_(n+1) - f_n)^2
= 2(f_n)^2 + 3(f_(n+1))^2 - ((f_(n+1))^2 - 2f_(n+1)f_n + (f_n)^2)
= 2(f_n)^2 + 3(f_(n+1))^2 - (f_(n+1))^2 + 2f_(n+1)f_n - (f_n)^2
= (f_n)^2 + 2(f_(n+1))^2 + 2f_(n+1)f_n
= (f_(n+1))^2 + (f_n)^2 + 2(f_n)f_(n+1) + (f_(n+1))^2
= (f_(n+1))^2 + (f_n + f_(n+1))^2
= (f_(n+1))^2 + (f_(n+2))^2
more readable. sorry...
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