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The Fibonacci sequence is an infinite sequence of numbers that have important consequences for theoretical mathematics...

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Advance Math

The Fibonacci sequence is an infinite sequence of numbers thathave important consequences for theoretical mathematics andapplications to arrangement of flower petals, population growth ofrabbits, and genetics. For each natural number n ? 1, the nthFibonacci number fn is defined inductively by

f1 = 1, f2 = 2, and fn+2 = fn+1 + fn

(a) Compute the first 8 Fibonacci numbers f1, · · · , f8.

(b) Show that for all natural numbers n, if ? = 1+?5 and ? =1??5, then fn = ?n??n .2 2 ???

Hint: Note the relationship between ? + 1 and ?2 and likewisebetween ? + 1 and ?2.

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