Why do the Lucas numbers use L1 equals 2 and L2 equals 1 and not L1 equals 1 and L2 equals 2 I have explain with logical reasoning and relevant calculations?

Answer 1

If L1=1 and L2=2, we would just get the Fibonacci sequence. Recall that the Fibonacci sequence is recursive and given by: f(0)=1, f(1)=1, and f(n)=f(n-1)+f(n-2) for integer n>1. Thus, we have f(2)=f(0)+f(1)=1+1=2. If L1=1 and L2=2 then we would have L1=f(1) and L2=f(2). Since the Lucas numbers are generated recursively just like the Fibonacci numbers, i.e. Ln=Ln-1+Ln-2 for n>2, we would have L3=L1+L2=f(1)+f(2)=f(3), L4=f(4), etc. You can use complete induction to show this for all n:

As we have already said, if L1=1 and L2=2, then we have L1=f(1) and L2=f(2). We now proceed to induction. Suppose for some m greater than or equal to 2 we have Ln=f(n) for n less than or equal to m. Then for m+1 we have, by definition, Lm+1=Lm+Lm-1. By the induction hypothesis, Lm+Lm-1=f(m)+f(m-1), but this is just f(m+1) by the definition of Fibonnaci numbers, i.e. Lm+1=f(m+1). So it follows that Ln=f(n) for all n if we let L1=1 and L2=2.