In the previous article, the relevant equation for multiple combinations (a and b) of the last and second last terms in the Fibonacci type sequence was x2 - ax - b = 0 and thereby unitary in the x2 term.
However we can generalize this first term also by considering another fascinating situation where we alternately vary the way in which we combine terms.
Thus is we alternately combine twice the last term with the previous term (a= 2, b = 1) , and then the last term with the previous (a = 1, b = 1), we generate the following sequence
0, 1, 1, 3, 4, 11, 15, 41, 56, 153, 209, 571, 780, ….
We now obtain two ratios when we divide a term by the previous term.
Thus the ratio of the last given terms in this series is 780/571 = 1.36602…
However starting with the previous term, the ratio is 571/209 = 2.73205…
Now the latter value approximates closely to the value of (the square root of 3) + 1, whereas the former value approximates to half of this
i.e. (the square root of 3 + 1)/2.
Now this value is the positive solution to the equation 2x2 - 2x - 1 = 0.
So when we combine terms alternately two and one at a time respectively, the coefficient of x2 will be equal to 2. Also both ratios generated (dividing the larger by the smaller) will themselves be in the same proportion as this coefficient of x2.
So in general terms when we alternately add multiple combinations of the last term (i. e. a > 1) with the previous term, and then combine the last with the previous, the solution for the smaller ratio generated will approximate to the positive solution for x to the equation ax2 - ax - b = 0.
Thus if we alternately combine 3 times the last with previous term, and then the last with the previous, we generate the sequence
0, 1, 1, 4, 5, 19, 23, 88, 111, 421, 532, 2017, 2549…
So the positive solution for x here is derived from the equation
3x2 - 3x - 1 = 0 i.e. (3 + square root of 21)/6 = 1.26376….
So the smaller ratio of this sequence will approximate to 1.26376…
The last two terms given 2549/2017 = 1.26375... which is a close approximation.
Also the value in turn of the larger divided by the lower ratio will closely approximate the coefficient of x2 which = 3.
2017/532 = 3.79135.. and 3.79135…/1.26375.. . is already very close to 3.
We can also combine alternately multiples of the second last term with the first terms and then the second last with the last.
For example if we alternately combine twice the second last with the last (a = 1, b= 2) and then the second last with the last (a = 1, b = 1), the following sequence emerges
0, 1, 1, 2, 4, 6, 14, 20, 48, 68, 164, 232, 560, 792…
The ratios are 792/560 = 1.4142... (which approximates the square root of 2)
The larger ratio 560/232 = 2.4137… approximates the (square root of 2) + 1.
This larger ratio approximates the positive solution for x in the general equation
x2 - bx - a = 0.
The difference of ratios is then given by b - a, which in this instance is 2 - 1 = 1.
I will illustrate this for the case where we alternately combine 4 times the second last term with the last (a = 1, b = 4) and then the second last with the last (a = 1, b = 1). This gives the sequence
0, 1, 1, 2, 6, 8, 32, 40, 168, 208, 880, 1088,…
The larger ratio will then approximate to the positive valued solution for x to the equation x2 - 4x - 1 = 0, which is 2 + square root of 5 = 4.236…
The difference in ratios will then be 4 - 1 = 3.
So from the terms given the larger ratio is 4.230… which approximates to 4.236...
The smaller ratio is 1088/880 = 1.236… which indeed differs from the larger
ratio by 3.