6. 10 pointsl The Lusqunczo is defined recursively s follows Lo=2 and Li=1 and a

Question

6. 10 pointsl The Lusqunczo is defined recursively s follows Lo=2 and Li=1 and and T,+1=Tn-1 +7, whenever n 1 This definition is very similar to the Fibonacci sequence2, but it starts with the initial terms Lo-2 and L, = l instead (a) Compute the first fifteen (15) Lucas numbers.3 (b) Let a denote the golden ratio, that is, let a denote the unique positive real number with the property 1.61803. Use a calculator to that a2-1 +a, we showed in homework 15 that = illustrate that Ln an-l for the values n = 1, 2, 3, 4, 5. (c) Use the principle of strong mathematical induction to prove that Ln a- for all integers n 2 1.

Explanation / Answer

Using the equation we get the following values for the first 15 Lucas numbers

alpha (n-1) values are as follows

We can see that Ln >= alpha (n-1) for L1,to L5

Strong mathematical induction

We have seen that the statement   Ln >= alpha (n-1) is true for n=1,2,3, 4 and 5

Hence by principle of strong induction, we can say that the statement is true for all n

n L 0 2 1 1 2 3 3 4 4 7 5 11 6 18 7 29 8 47 9 76 10 123 11 199 12 322 13 521 14 843

alpha (n-1) values are as follows

1 1 1 2 3 1.61803 3 4 2.618021 4 7 4.236037 5 11 6.854034

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