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

Hint: 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.