The Fibonacci sequence is defined by

Question:

The Fibonacci sequence is defined by

a1 = 1 = a2an = an − 1 + an − 2 for > 2

Find $\frac{a_{n+1}}{a_{n}}$ for $n=1,2,3,4,5$.

Solution:

$a_{1}=1=a_{2}, a_{n}=a_{n-1}+a_{n-2}$ for $n>2$

Then we have:

$a_{3}=a_{2}+a_{1}=1+1=2$

$a_{4}=a_{3}+a_{2}=2+1=3$

$a_{5}=a_{4}+a_{3}=3+2=5$

$a_{6}=a_{5}+a_{4}=5+3=8$

For $n=1, \frac{a_{n+1}}{a_{n}}=\frac{a_{2}}{a_{1}}=\frac{1}{1}=1$

For $n=2, \frac{a_{n+1}}{a_{n}}=\frac{a_{3}}{a_{2}}=\frac{2}{1}=2$

For $n=3, \frac{a_{n+1}}{a_{n}}=\frac{a_{4}}{a_{3}}=\frac{3}{2}$

For $n=4, \frac{a_{n+1}}{a_{n}}=\frac{a_{5}}{a_{4}}=\frac{5}{3}$

For $n=5, \frac{a_{n+1}}{a_{n}}=\frac{a_{6}}{a_{5}}=\frac{8}{5}$

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