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

The Fibonacci sequence is an infinite sequence of numbers that have important consequences for theoretical mathematics and applications to arrangement of flower petals, population growth of rabbits, and genetics. For each natural number n ≥ 1, the nth Fibonacci 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 α = 1+√5 and β = 1−√5, then fn = αn−βn .2 2 α−β

Hint: Note the relationship between α + 1 and α2 and likewise between β + 1 and β2.

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Expert Solution

Colby Messinger answered 1 year ago

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