The fibonacci sequence is defined by a_1=1=a_2, a_n=a_ n-1 +a_ n-2 from n>2. Find a_ n+1 a_n for n=1, 2, 3, 4, 5.

a_n=a_ n-1 +a_ n-2 For n>2 a_3= a_ 3-1 + a_ 3-2 = a_2+ a_1=1+1=2 a_4= a_ 4-1 + a_ 4-2 = a_3+ a_2=2+1=3 a_5= a_ 5-1 + a_ 5-2 =a_4+ a_3=3+2=3 a_6= a_ 6-1 + a_ 6-2 =a_5+ a_4=5+31=8 for n=1 a_ n-1 a _n = a_2 a_1 =1 1 =1 For n=2 a^3 a^2 =2 1 =2 For n=3 a^3 a^2 =3 2 =1.5 For n=4 and n=5 a_5 a_4 =5 3 and a_6 5_5 =8 5 The reqquired series is 1,2,3 2 ,5 3 ,8 5 ,...

The fibonacci sequence is defined by a_1=1=a_2, a_n=a_ n-1 +a_ n-…

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Question

The fibonacci sequence is defined by $\text{a}_1=1=\text{a}_2,\ \text{a}_\text{n}=\text{a}_{\text{n}-1}+\text{a}_{\text{n}-2}$ from $\text{n}>2.$ Find $\frac{\text{a}_{\text{n}+1}}{\text{a}_\text{n}}$ for $\text{n}=1,\ 2,\ 3,\ 4,\ 5.$

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Answer

$\text{a}_\text{n}=\text{a}_{\text{n}-1}+\text{a}_{\text{n}-2}$ For $\text{n}>2$
$\Rightarrow\text{a}_3=\text{a}_{3-1}+\text{a}_{3-2}=\text{a}_2+\text{a}_1=1+1=2$
$\Rightarrow\text{a}_4=\text{a}_{4-1}+\text{a}_{4-2}=\text{a}_3+\text{a}_2=2+1=3$
$\Rightarrow\text{a}_5=\text{a}_{5-1}+\text{a}_{5-2}=\text{a}_4+\text{a}_3=3+2=3$
$\Rightarrow\text{a}_6=\text{a}_{6-1}+\text{a}_{6-2}=\text{a}_5+\text{a}_4=5+31=8$
$\therefore$ for $\text{n}=1$
$\frac{\text{a}_{\text{n}-1}}{\text{a}_\text{n}}=\frac{\text{a}_2}{\text{a}_1}=\frac{1}{1}=1$
For $\text{n}=2$
$\frac{\text{a}^3}{\text{a}^2}=\frac{2}{1}=2$
For $\text{n}=3$
$\frac{\text{a}^3}{\text{a}^2}=\frac{3}{2}=1.5$
For $\text{n}=4$ and $\text{n}=5$
$\frac{\text{a}_5}{\text{a}_4}=\frac{5}{3}$ and $\frac{\text{a}_6}{\text{5}_5}=\frac{8}{5}$
$\therefore$ The reqquired series is $1,2,\frac{3}{2},\frac{5}{3},\frac{8}{5},...$

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