= f6 .
IV. The sum of the first n Fibonacci numbers with even indices is
f2 + f4 + ... + f2n = f2n+1 - 1.
Proof:
From II,
f1 + f2 + f3 + ... + f2n = f2n+2 - 1.
When we subtract the result from III, we get the desired result.
Example:
f2 + f4 + f6 + f8 + f10 + f12 = 1 +3 + 8 + 21 +55 +144
= 232
= f13 - 1 .
V. The sum of all (fn+1)/ (fn )
converges to the Golden Ratio.
3/1 + 5/3 + 8/5 + 13/8 .... converges to ) / 2.
Proof that Rn converges to the Golden Ratio:
Let R = lim Rn as n approaches infinity
= lim f n+1 / f n as n approaches infinity
= lim fn + fn-1 /fn as n approaches infinity
= lim (1 + fn-1/ fn) as n approaches infinity
= 1 + lim (fn-1 /fn ) as n approaches infinity
= 1 + 1/ lim (fn-1 /fn ) as n approaches infinity
= 1 + 1/R
So, R = 1 + 1/R,
or R^2 = R + 1,
R^2 - R - 1 = 0,
R = ( ) / 2
= ( ) / 2
Since Rn is positive, Rn = ( ) / 2 .
Thus, Rn converges to the Golden Ratio.