Consider the identity, 𝑓𝑚+𝑛 = 𝑓𝑚𝑓𝑛+1 + 𝑓𝑛𝑓𝑚−1, where 𝑓𝑖 is the i-th Fibonacci number.

a. Evaluate this identity for some instances of positive integers 𝑚 and 𝑛.
b. Find the expression when 𝑚 = 𝑛.
c. Using the previous expression found in part b, prove that 𝑓𝑚 and 𝑓2𝑚. In other words, prove that 𝑓2𝑚/𝑓𝑚 is an integer. Call that integer 𝑋𝑚.
d. Find the values of 𝑋1,𝑋2,𝑋3,𝑋4, 𝑋5, 𝑋6.
e. Do you think you can find (not prove) a recurrence formula for 𝑋𝑖 ?

Did you know?

Did you know that there is an interesting identity related to the Fibonacci numbers? The identity is 𝑓𝑚+𝑛 = 𝑓𝑚𝑓𝑛+1 + 𝑓𝑛𝑓𝑚−1, where 𝑓𝑖 is the i-th Fibonacci number. a. We can evaluate this identity for different instances of positive integers 𝑚 and 𝑛, which can give us some insights into the relationship between the Fibonacci numbers. b. When 𝑚 = 𝑛, the expression simplifies further. You can find the simplified expression by substituting 𝑚 = 𝑛 in the original identity. c. Using the simplified expression found in part b, we can prove that 𝑓𝑚 and 𝑓2𝑚 have a special relationship. In other words, we can prove that 𝑓2𝑚 divided by 𝑓𝑚 is always an integer. Let's call that integer 𝑋𝑚. d. By finding multiple instances of 𝑋𝑚 for different values of 𝑚 (such as 𝑋1, 𝑋2, 𝑋3, 𝑋4, 𝑋5, 𝑋6), we can see if there is any pattern or recurring sequence for these values. e. As an interesting challenge, you can try to find a recurrence formula for 𝑋𝑖. While it is not necessary to prove the formula, it can be an intriguing exercise to see if there is a pattern in the values of 𝑋𝑖 as 𝑖 increases.