The first two terms of the Fibonacci sequence are 1 1 Find t

The first two terms of the Fibonacci sequence are 1, 1. Find two other integers that, together with the recurrence a_n = a_n - 1+a_n - 2, do not generate Fibonacci numbers. And, find two other integers that, together with the recurrence a_n = a_n-1 +a_n-2, do generate Fibonacci numbers. What kinds of integers do this?

Solution

We let first two terms be 1,3 because in Fibonacci sequence after 1 we have 2

So the sequence it generates is

1,3,4,7,11,18,29,47,... which is not the Fibonnaci sequence

To generate Fibonacci numbers with other integers we take any pair of consecutive intgers in Fibonacci sequence eg. , 2,3 as starting terms

So we get

2,3,5,8,13 and so on which the Fibonacci sequence from 2 onwards

Any consecutive pair of integers from Fibonacci sequence can be taken as starting numbers

For generate a sequence take any consecutive pair which is not in Fibonacci sequence.