The Fibonacci sequence is defined by 1 = a_1 = a_2 and a_n = a_ n…

Question

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

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Answer

Given, $1=a_1=a_2$
and $a_n=a_{n-1}+a_{n-2}, n>2$
On putting $n=3,4,5,6$ respectively, we get
For $n=3, a_3=a_{3-1}+a_{3-2}=a_2+a_1=1+1=2$
For $n=4, a_4=a_{4-1}+a_{4-2}=a_3+a_2=2+1=3$
For $\mathrm{n}=5, \mathrm{a}_5=\mathrm{a}_{5-1}+\mathrm{a}_{5-2}=\mathrm{a}_4+\mathrm{a}_3=3+2=5$
For $n=6, a_6=a_{6-1}+a_{6-2}=a_5+a_4=5+3=8$
Now, $\frac{a_{n+1}}{a_n}$, for $n=1,2,3,4,5$.
For n = 1, $\frac { a _ { 2 } } { a _ { 1 } } = \frac { 1 } { 1 }$ = 1
For n = 2, $\frac { a _ { 3 } } { a _ { 2 } } = \frac { 2 } { 1 }$ = 2
For n = 3, $\frac { a _ { 4 } } { a _ { 3 } } = \frac { 3 } { 2 }$
For n = 4, $\frac { a _ { 5 } } { a _ { 4 } } = \frac { 5 } { 3 }$
For n = 5, $\frac { a _ { 6 } } { a _ { 5 } } = \frac { 8 } { 5 }$
Hence, the terms are 1, 2, $\frac { 3 } { 2 } , \frac { 5 } { 3 }$ and $\frac { 8 } { 5 }$

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