Question

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 $n=1,2,3,4,5.$

• A. $1,2,\frac{3}{2}$ $,\frac{5}{3}$ and $\frac{8}{5}$
• B. $1,2,\frac{3}{2},4$ and $8$
• C. $3,4,\frac{9}{2}$, $5$ and $8$
• D. $1,2,\frac{3}{2}$ $6$ and $9$

Answer 1

Answer: (A)

$1,2,\frac{3}{2}$ $,\frac{5}{3}$ and $\frac{8}{5}$

According to the question $a_1=a_2=1$and $a_n=a_{n-1}+a_{n-2}$
Substituting $n=3,4,5$ and $6$,
$a_3=a_2+a_1=1+1=2$
$a_4=a_3+a_2=2+1=3$
$a_5=a_4+a_3=3+2=5$
and $a_6=a_5+a_4=5+3=8$

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

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

$n=3$, $\frac{a_4}{a_3}=\frac{3}{2}$

$n=4$, $\frac{a_5}{a_4}=\frac{5}{3}$

$n=5$, $\frac{a_6}{a_5}=\frac{8}{5}$

Hence, option $A$ is correct.