Question

Let a sequence be defined by $a_1=1,a_2=1$ and, $a_n=a_{n-1}+a_{n-2}$ for all $n>2$,
find $\frac{a_{n+1}}{a_n}$ for $n=1,2,3,4$.

Answer 1

We have, $a_1=1,a_2=1$
and, $a_n=a_{n-1}+a_{n-2}$ for all $n>2$

Putting $n=3,4$ and $5$, we get

$a_3=a_2+a_1=1+1=2$
$a_4=a_3+a_2=2+1=3$
and, $a_5=a_4+a_3=2+1=5$

Thus, we have
$a_1=1,a_2=1,a_3=2,a_4=3$ and $a_5=5$

Now, putting $n=1,2,3$ and $4$ in $\frac{a_{n+1}}{a_n}$, we get
$\frac{a_2}{a_1}=\frac{1}{1}=1$ $[\because a_1=a_2=1]$
$\frac{a_3}{a_2}=\frac{2}{1}=2$ $[\because a_2=1$and $a_3=2]$
$\frac{a_4}{a_3}=\frac{3}{2}$ $[\because a_4=3$ and $a_3=2]$
$\frac{a_5}{a_4}=\frac{5}{3}$ $[\because a_4=3$ and $a_5=5]$