Fibonacci sequences

Joshua DuMont

The Fibonacci sequence is defined by fn = fn-1+ fn-2, f0 = 1, f1= 1. We could just as easily talk about the sequence where we add the two previous terms to get the next term and start with different initial conditions. A table giving the first 30 terms of three of these sequences is shown along with the ratios of successive terms.

 n fn gn hn 0 0 4 5 1 1 7 34 2 1 1 11 1.57142857 39 1.14705882 3 2 2 18 1.63636364 73 1.87179487 4 3 1.5 29 1.61111111 112 1.53424658 5 5 1.66666667 47 1.62068966 185 1.65178571 6 8 1.6 76 1.61702128 297 1.60540541 7 13 1.625 123 1.61842105 482 1.62289562 8 21 1.61538462 199 1.61788618 779 1.61618257 9 34 1.61904762 322 1.61809045 1261 1.61874198 10 55 1.61764706 521 1.61801242 2040 1.61776368 11 89 1.61818182 843 1.61804223 3301 1.61813725 12 144 1.61797753 1364 1.61803084 5341 1.61799455 13 233 1.61805556 2207 1.61803519 8642 1.61804905 14 377 1.61802575 3571 1.61803353 13983 1.61802823 15 610 1.61803714 5778 1.61803416 22625 1.61803619 16 987 1.61803279 9349 1.61803392 36608 1.61803315 17 1597 1.61803445 15127 1.61803401 59233 1.61803431 18 2584 1.61803381 24476 1.61803398 95841 1.61803387 19 4181 1.61803406 39603 1.61803399 155074 1.61803404 20 6765 1.61803396 64079 1.61803399 250915 1.61803397 21 10946 1.618034 103682 1.61803399 405989 1.618034 22 17711 1.61803399 167761 1.61803399 656904 1.61803399 23 28657 1.61803399 271443 1.61803399 1062893 1.61803399 24 46368 1.61803399 439204 1.61803399 1719797 1.61803399 25 75025 1.61803399 710647 1.61803399 2782690 1.61803399 26 121393 1.61803399 1149851 1.61803399 4502487 1.61803399 27 196418 1.61803399 1860498 1.61803399 7285177 1.61803399 28 317811 1.61803399 3010349 1.61803399 11787664 1.61803399 29 514229 1.61803399 4870847 1.61803399 19072841 1.61803399

Notice that the ratio of a term to the previous term seems to approach a particular value, and that that value is the same for all three sets of initial conditions.

Define

This means = Φ since we have our recursive formula for fn.

We can rearrange and say that as :

fn-1+fn-2= Φ *fn-1

fn-2= Φ *fn-1- fn-1

fn-2=( Φ - 1)*fn-1

=fn-1

=

As , =.

We can conclude that:

Φ =

Φ *( Φ - 1)=1

Φ 2- Φ -1=0

Φ =  ≈1.618

{Note Φ = is unreasonable as the terms in sequence will be positive}

This process did not involve the initial conditions f0 and f1, so it will be the same for different choices for the first two terms.

We could also consider ratios of every second term or third term etc.

 n fn 0 0 1 1 2 1 1 3 2 2 2 4 3 1.5 3 3 5 5 1.66666667 2.5 5 5 6 8 1.6 2.66666667 4 8 7 13 1.625 2.6 4.33333333 6.5 8 21 1.61538462 2.625 4.2 7 9 34 1.61904762 2.61538462 4.25 6.8 10 55 1.61764706 2.61904762 4.23076923 6.875 11 89 1.61818182 2.61764706 4.23809524 6.84615385 12 144 1.61797753 2.61818182 4.23529412 6.85714286 13 233 1.61805556 2.61797753 4.23636364 6.85294118 14 377 1.61802575 2.61805556 4.23595506 6.85454545 15 610 1.61803714 2.61802575 4.23611111 6.85393258 16 987 1.61803279 2.61803714 4.2360515 6.85416667 17 1597 1.61803445 2.61803279 4.23607427 6.85407725 18 2584 1.61803381 2.61803445 4.23606557 6.85411141 19 4181 1.61803406 2.61803381 4.2360689 6.85409836 20 6765 1.61803396 2.61803406 4.23606763 6.85410334 21 10946 1.618034 2.61803396 4.23606811 6.85410144 22 17711 1.61803399 2.618034 4.23606793 6.85410217 23 28657 1.61803399 2.61803399 4.236068 6.85410189 24 46368 1.61803399 2.61803399 4.23606797 6.854102 25 75025 1.61803399 2.61803399 4.23606798 6.85410196 26 121393 1.61803399 2.61803399 4.23606798 6.85410197 27 196418 1.61803399 2.61803399 4.23606798 6.85410196 28 317811 1.61803399 2.61803399 4.23606798 6.85410197 29 514229 1.61803399 2.61803399 4.23606798 6.85410197

These also seem to have limits. More than that it seems that the limits have the same pattern as the Fibonacci sequence itself. We guess that:

=+

By our recursive formula for fn, =

Since the values n-1 and n-k, and the values n and n-(k-1) are the same distance apart we can say that in the limits:

=.

= for similar reasons.

So, =+  as we guessed.