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Sequence

The Fibonacci...

Question

The Fibonacci sequence is defined by $1 = a_{1} = a_{2}$ and $a_{n} = a_{n - 1} + a_{n - 2}, n > 2$
Find $\frac{a_{n + 1}}{a_{n}}, f o r n = 1, 2, 3, 4, 5$

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Solution

$1 = a_{1} = a_{2} a_{n} = a_{n - 1} + a_{n - 2}, n > 2 ∴ a_{3} = a_{2} + a_{1} = 1 + 1 = 2 a_{4} = a_{3} + a_{2} = 2 + 1 = 3 a_{5} = a_{4} + a_{3} = 3 + 2 = 5 a_{6} = a_{5} + a_{4} = 5 + 3 = 8$
$∴ F o r n = 1, \frac{a_{n} + 1}{a_{n}} = \frac{a_{2}}{a_{1}} = \frac{1}{1} = 1 F o r n = 2, \frac{a_{n} + 1}{a_{n}} = \frac{a_{3}}{a_{2}} = \frac{2}{1} = 2 F o r n = 3, \frac{a_{n} + 1}{a_{n}} = \frac{a_{4}}{a_{3}} = \frac{3}{2} F o r n = 4, \frac{a_{n} + 1}{a_{n}} = \frac{a_{5}}{a_{4}} = \frac{5}{3} F o r n = 5, \frac{a_{n} + 1}{a_{n}} = \frac{a_{6}}{a_{5}} = \frac{8}{5}$

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