Question

The Fibonacci sequence is defined by $a_1=a_2=1$ and $a_n=a_{n-1}+a_{n-2},n>2$. Then, match the terms and ratios given in Column I with their values given in Column II and choose the correct option from the codes given below.

Column I		Column II
A	$a_1,a_2,a_3$	1	$2,3,5$
B	$a_3,a_4,a_5$	2	$\frac{3}{2},\frac{5}{3},\frac{8}{5}$
C	$\frac{a_4}{a_3},\frac{a_5}{a_4},\frac{a_6}{a_5}$	3	$1,2,\frac{2}{3}$
D	$\frac{a_2}{a_1},\frac{a_3}{a_2},\frac{a_3}{a_4}$	4	$1,1,2$

• A. A - (4), B - (1), C -(3), D -(2)
• B. A - (4), B - (1), C -(2), D -(3)
• C. A - (1), B - (4), C -(3), D -(2)
• D. A - (1), B - (4), C -(2), D -(3)

Answer 1

Answer: (B)

Here, $1=a_1=a_2,a_n=a_{n-1}+a_{n-2},n>2$
Putting $n=3,4,5,6$,
At $n=3$, $a_3=a_{3-1}+a_{3-2}$
$=a_2+a_1$
$=1+1=2$
At $n=4$, $a_4=a_{4-1}+a_{4-2}$
$=a_3+a_2$
$=2+1=3$
At $n=5$, $a_5=a_{5-1}+a_{5-2}$
$=a_4+a_3=3+2=5$
At $n=6$, $a_6=a_{6-1}+a_{6-2}$
$=a_5+a_4=5+3=8$
Now, $\frac{a_{n+1}}{a_n}$, for $n=1,2,34,5$.
At $n=1$, $\frac{a_2}{a_1}=\frac{1}{1}=1$
At $n=2$, $\frac{a_3}{a_2}=\frac{2}{1}=2$
At $n=3$, $\frac{a_4}{a_3}=\frac{3}{2}$
At $n=4$, $\frac{a_5}{a_4}=\frac{5}{3}$
At $n=5$, $\frac{a_6}{a_5}=\frac{8}{5}$
Hence, the terms are $1,2,\frac{3}{2},\frac{5}{3}$ and $\frac{8}{5}$.