Question

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

Answer 1

Finding sequence.

Given, $a_1=a_2=1$ and $a_n=a_{n-1}+a_{n-2}\forall n>2,nϵN$.

Substituting $n=3,4,5$ and $6$ in $a_n$, we get

$a_3=a_{3-1}+a_{3-2}$
$\Rightarrow a_3=a_2+a_1$
$\Rightarrow a_3=1+1$
$\Rightarrow a_3=2$

$a_4=a_{4-1}+a_{4-2}$
$\Rightarrow a_4=a_3+a_2$
$\Rightarrow a_4=2+1$
$\Rightarrow a_4=3$

$a_5=a_{5-1}+a_{5-2}$
$\Rightarrow a_5=a_4+a_3$
$\Rightarrow a_5=3+2$
$\Rightarrow a_5=5$

$a_6=a_{6-1}+a_{6-2}$
$\Rightarrow a_6=a_5+a_4$
$\Rightarrow a_6=5+3$
$\Rightarrow a_6=8$

Thus, the Fibonacci sequence is $1,1,2,3,5,8,...$.

Finding value of $\frac{a_{n+1}}{a_n}$ for $n=1,2,3,4,5$
Now, for $n=1$,
$\frac{a_{1+1}}{a_1}=\frac{a_2}{a_1}=\frac{1}{1}=1$

for $n=2$,
$\frac{a_{2+1}}{a_2}=\frac{a_3}{a_2}=\frac{2}{1}=2$

for $n=3$
$\frac{a_{3+1}}{a_3}=\frac{a_4}{a_3}=\frac{3}{2}$

for $n=4$
$\frac{a_{4+1}}{a_4}=\frac{a_5}{a_4}=\frac{5}{3}$

for $n=5$
$\frac{a_{5+1}}{a_5}=\frac{a_6}{a_5}=\frac{8}{5}$

Final answer:
Hence, the Fibonacci sequence is $1,1,2,3,5,8,...$ and the values of $\frac{a_{n+1}}{a_n}$ for $n=1,2,3,4\text{and}5$ are $1,2,\frac{3}{2},\frac{5}{3}$ and $\frac{8}{5}$ respectively.