Once again the 3-term Fibonacci type sequence is

0, 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, ….(This is obtained by successively combining the last 3-terms to obtain the next term in the sequence to obtain the next term).

The common ratio of k/(k + 1) where k represents the kth term approximates to 1.839286755…

I refer to this ratio (of the

3-termsequence) asPhi_{3}

As we have seen

Phirepresents the real (positive- valued) solution for x to the cubic equation_{3 }

x;^{3}- x^{2 }- x - 1 = 0The solution to this equation can be expressed as follows:

x = Phi_{3 }= 1/3 + {19/27 + (11/27)^{1/2}}^{1/3}+ {19/27 - (11/27)^{1/2}}^{1/3}There would appear to be a link here with the actual terms in the original sequence

To do this we express we multiply the fractional terms inside the bracket by 3/3 so that

x = Phi_{3 }= 1/3 + {57/81 + (33/81)^{1/2}}^{1/3}+ {57/81 - (33/81)^{1/2}}^{1/3}The denominator of the terms (inside both sets of brackets) is

81 = Tin the sequence._{11 }Also the difference of numerators (inside both sets of brackets)

= 57 - 33 = 24 =T_{9};In the

2-termcasefor theFibonacci Sequence, Phirepresents the positive solution for x to the quadratic equation_{2 }

x;^{2 }- x - 1 = 0

x=Phi_{2 }(= Phi) = 1/2 + (5^{1/2})/2

Thus whereas

thetwo-termratio= 1/2 + a square root expression,thethree-termratio=

1/3 +a cube root expression