Question

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

Answer 1

We need to find $n=1, 2, 3,4,5$ separately as the question says. Fibonacci sequence is the series of numbers in which a number is the addition of the last two numbers starting with $0$ or $1$ . Fibonacci sequence is significant and used to create technical indicators.

Complete answer:
Given $a{_1}= \ 1$
$a{_2}= \ 1$
We need to find $a{_3}$ , $a{_4}$ , $a{_5}$ and $a{_6}$
Given,
$a{_n}\ = \ a{_{n-1}}+ \ a{_{n-2}}$ for $\ n > 2$
Now we need to find for $n = 3,$
$a{_3} = \ a{_{3-1}} + \ a{_{3-2}} = \ a{_2}+ \ a{_1}$
= $\ 1 + 1$
By adding ,
= $\ 2$
Then need to find for $n = \ 4$ ,
$a{_4}= \ a{_{4-1}}+ \ a{_{4-2}} =\ a{_3}+ \ a{_2}$
= $\ \ 2\ + \ 1$
By adding,
= $\ 3$
Next for $n\ = 5,$
$a{_5}= \ a{_{5-1}}+ \ a{_{5-2}} = a{_4}+ \ a{_3}$
= $\ 3 + 2$
By adding,
We get,
= $\ 5$
Finally need to find for $\ n\ = 6$
$a{_6} = \ a{_{6-1}}+ \ a{_{6-2}} =\ a{_5}+ \ a{_4}$
= $5 + 3$
By adding,
We get,
= $8$
Also given,
$a{_n}= \ a{_{n- 1}}+ a{_{n-2}}$
Now here we need to find for $n = \ 1,$
$(\ a{_{n+1}}\ /a{_n})= \ (a{_{1+1}}\ /a{_1} ) =\ (\ a{_2}/a{_1})$
By substituting the values,
We get,
= $\dfrac{1}{1}$
= $\ 1$
Then for $n = 2$
$(a{_{n+1}}/a{_n})= \ ( a{_{2+1}}/\ a{_2})$
$=\ (a{_3}/\ a{_2})$
By substituting the values,
We get,
= $\dfrac{2}{1}$
= $\ 2$
Now for $n = 3$
$(a{_{n+1}} /\ a{_n})$ $= \ (\ a{_{3+1}}/a{_3} )$
$= \ (a{_4}/\ a{_3})$
By substituting the values,
We get,
= $\dfrac{3}{2}$
Then need to find for $n = 4$
$(a{_{n+1}}/\ a{_n})$ $= \ (a{_{4+1}}/$ $a{_4})= (a{_5}/a{_4})$
By substituting the values,
We get,
= $\dfrac{5}{3}$
Finally for $n = 5$
$(a{_{n+1}}/\ a{_n})$
= $(\ a{_{5+1}}/\ a{_5})$
= $\ (a{_6}/\ a{_5})$
By substituting the values,
We get,
= $\dfrac{8}{5}$
Hence the value of $(a{_{n+1}}/\ a{_n})$ when n\ = \ 1,2,3,4,5\ are $1,2,\dfrac{3}{2},\dfrac{5}{3},\dfrac{8}{5}$ respectively.
Final answer :
The value of $(a{_{n+1}}/\ a{_n})$ when n\ = \ 1,2,3,4,5\ are $1,2,\dfrac{3}{2},\dfrac{5}{3},\dfrac{8}{5}$ respectively.

Note:
Another example of Fibonacci sequence is $0,1,1,2,3,5,8,13,21,....$ The expression of the Fibonacci sequence is $X{_n} = X{_{n-1}} + X{_{n-2}}$ . Mathematically Fibonacci number is strongly related to golden ratio and also closely related to lucas numbers.