Proof that the formula Fn = Fn-1 + Fn-2 works
Fibonacci python

Umm— fn-1+Fn-2?? Fibonacci python

= 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.