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 ?