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# The Fibanocci sequence is defined by $1=a_1=a_2\:\:and\:\:a_n=a_{n-1}+a_{n-2}\:\:for\:n>2$ Find $\large\frac{a_{n+1}}{a_n}$, for $n=1,2,3,4,5$

$\begin{array}{1 1}1, 2, 3/2, 5/3, 9/5 \\1, 2, 3/2, 4/3, 8/5 \\1, 2, 5/2, 5/3, 8/5 \\ 1, 2, 3/2, 5/3, 8/5 \end{array}$

Given $a_n=a_{n-1}+a_{n-2}$ and $a_1=a_2=1$
By putting $n=3,4,......$ we get
$a_3=a_2+a_1=1+1=2$
$a_4=a_3+a_2=2+1=3$
$a_5=a_4+a_3=3+2=5$
$a_6=a_5+a_4=5+3=8$
Similarly by putting $n=1,2,....5$ in $\large\frac{a_{n+1}}{a_n}$ we get
$for\:\:n=1\:\:\large\frac{a_{n+1}}{a_n}=\frac{a_2}{a_1}$$=1 for\:\:n=2\:\:\large\frac{a_{n+1}}{a_n}=\frac{a_3}{a_2}=\frac{2}{1}$$=2$
$for\:\:n=3\:\:\large\frac{a_{n+1}}{a_n}=\frac{a_4}{a_3}=\frac{3}{2}$
$for\:\:n=4\:\:\large\frac{a_{n+1}}{a_n}=\frac{a_5}{a_4}=\frac{5}{3}$
$for\:\:n=5\:\:\large\frac{a_{n+1}}{a_n}=\frac{a_5}{a_5}=\frac{8}{5}$