Question

Suppose we build a sequence of numbers using the method of adding the previous two numbers to build the next one. this time, however, suppose our first two numbers are 2 and 1. generate the first 15 terms. this sequence is called the lucas sequence and is written as l1, l2, l3 , …. compute the quotients of consecutive terms of the lucas sequence as we did with the fibonacci numbers. what number do these quotients approach? what role do the initial values play in determining what number the quotients approach? try two other first terms and generate a sequence. what do the quotients approach?

Answer 1

2 , 1,3,4,7,11,18,29,47,76,123,199,322,521,843

quotients  are 0.5 ,2,1.333,1.571,1.636,1.611, 1.621,1.617, 1.6184,1.6179,1.6181,1.6180,1.61801, 1.61804

The ratios are approaching the Golden Number.

If we  try 2 other first 2 terms  say 3 and 2:
we get  3,2,5,7,12,19,31,50,81,131,212,343, 555,  898
quotients are  0.67,2.5,1.4,1.714,1.5833,1.631 , 1.612, 1.62,1.617, 1.6183
,1.6181,1.61801   Again we are approaching the  golden number