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Fibonacci Series

The Fibonacci...

Question

The Fibonacci sequence is defined by
$1 = a_{1} = a_{2} n d a_{n} = a_{n - 1} + a_{n - 1}, n > 2$ .

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

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Solution

Given : $1 = a_{1} = a_{2} a n d a_{n} = a_{n - 1} + a_{n - 1}, n > 2$ Find $\frac{a_{n + 1}}{a_{n}} f o r n = 1, 2, 3, 4, 5 .$
Substituting n = 3 , 4 , 5 , 6 we get
$a_{3} = a_{2} + a_{1} = 1 + 1 = 2$
$a_{4} = a_{3} + a_{2} = 2 + 1 = 3$
$a_{5} = a_{4} + a_{3} = 3 + 2 = 5$
$a_{6} = a_{5} + a_{4} = 5 + 3 = 8$
To find ; $\frac{a_{n + 1}}{a_{n}}, n = 1, 2, 3, 4, 5$
For $n = 1 \frac{a_{n + 1}}{a_{n}} = \frac{a_{2}}{a_{1}} = \frac{1}{1} = 1$
For $n = 2, \frac{a_{n + 1}}{a_{n}} = \frac{a_{3}}{a_{2}} = \frac{2}{1} = 2$

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