In the Fibonacci Sequence

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,…

When we take any four terms in the sequence such as 1, 1, 2, 3 the product of the first and last and twice the product of the inner two give integral solutions for the adjacent and opposite sides of the Pythagorean (right-angled) triangle.

Thus in this case

3* 1 =3and2(1 * 2)=4are the relevant sides.Also the hypotenuse i.e. in this case

5is a term in the Fibonacci sequence.A similar type relationship holds for the Lucas series

1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 501,…Again when we take four terms in the sequence such as

1, 3, 4, 7the product of the first and last and twice the product of the inner two, also give two integral solutions for the adjacent and opposite sides of the Pythagorean (right-angled) triangle.Thus in this case

7 * 1 = 7and2(3 * 4) =24are the relevant sides.In this case the hypotenuse is the sum of two terms,

Tand_{k }Tin the Lucas Series._{k + 2 }In this case the hypotenuse =

T=_{4 }+ T_{6 }7 + 18=25Here

Tis_{4 }the last_{ }term of the 4 term sequence._{ }When

Tis the last term in the four-term sequence, the hypotenuse =_{5 }T_{6 }+ T_{8};When

Tis the last term in the four-term sequence, the hypotenuse =_{6 }T_{8 }+ T_{10 }etcSo we moving out one term with each subsequent four-term sequence to drive the relevant hypotenuse terms.

Thus when

Tis the last term in the four-term sequence, the hypotenuse =_{7 }T_{10 }+ T_{12}The four-term sequence =

7, 11, 18, 29Therefore the adjacent side =

29 * 7=203;The opposite side =

2(11 * 18)=396The hypotenuse

123 + 322 =445Incidentally, when one compares the product of the last two terms in a four-term sequence with the product of the inner two, and interesting result emerges in that the difference always has an absolute value of

5.Thus with the first four, the outer product = 7 and the inner product = 12 so that the difference =

5.Likewise with

7, 11, 18, 29the outer product =203and the inner product =198again with a difference =5; (the sign alternates with each successive sequence!)What is remarkable is that the hypotenuse generated in the Lucas case is always divisble by

5with the result representing a term in the Fobonacci sequence.

Thus for example445=89*5.(Interestingly there is a fascinating connection here with the sum of the Fibonacci and Lucas decimal point sequences. As we know the sum of the point series for the Fibonacci terms = the reciprocal of89whereas the sum of the Lucas point terms = the reciprocal of (89*5)/6).In the Fibonacci Sequence when we apply the same procedure the difference always has an absolute value of

1.Thus for example with the first four-term sequence, the outer product = 3 and the inner product = 2 with a difference =

1.