Question

The Fibonacci sequence is defined by 1 = a1 = a2 and an = an – 1 + an – 2, n > 2. Find an + $\frac{1}{an}$ , for n = 1, 2, 3, 4, 5.

Answer 1

1 = a1 = a2

an = an -1 + an - 2, n >2

∴ a3 = a2 + a1 = 1 + 1 =2

a4 = a3 + a2 = 2 + 1 = 3

a5 = a4 + a3 = 3 + 2 = 5

a6 = a5 + a4 = 5 + 3 = 8

∴ For n = 1, an + $\frac{1}{an}=\frac{a2}{a1}=\frac{1}{1}$

For n = 2 , an + $\frac{1}{an}=\frac{a3}{a2}=\frac{2}{1}$

For n = 3 , an + $\frac{1}{an}=\frac{a4}{a3}=\frac{3}{2}$

For n = 4 , $\frac{1}{an}=\frac{a5}{a4}=\frac{5}{3}$

For n = 5 , an + $\frac{1}{an}=\frac{a6}{a5}=\frac{8}{5}$.