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

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Solution

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

$\Rightarrow a_{3} = a_{3 - 1} + a_{3 - 2} = a_{2} + a_{1} = 1 + 1 = 2$

$\Rightarrow a_{4} = a_{4 - 1} + a_{4 - 2} = a_{3} + a_{2} = 2 + 1 = 3$

$\Rightarrow a_{5} = a_{5 - 1} + a_{5 - 2} = a_{4} + a_{3} = 3 + 2 = 5$

$\Rightarrow a_{6} = a_{6 - 1} + a_{6 - 2} = a_{5} + a_{4} = 5 + 3 = 8$

$∴$ For n = 1

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

For n = 2

$\frac{a_{3}}{a_{2}} = \frac{2}{1} = 2$

For n = 3

$\frac{a_{4}}{a_{3}} = \frac{3}{2} = 1.5$

For n = 4 and n = 5

$\frac{a_{5}}{a_{4}} = \frac{5}{3}$ and $\frac{a_{6}}{a_{5}} = \frac{8}{5}$

$∴$ The required series is $1, 2, \frac{3}{2}, \frac{5}{3}, \frac{8}{5}, \dots \dots$

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