Question

The fibonacci sequence is defined by 1=a1=a2 and an=an-1 + an-2,n>2.
Find an+1/an for n=1,2,3,4,5.

Solve it in a copy.

Answer 1

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Answer 2

$\Large{\textbf{\underline{\underline{According\:to\:the\:Question}}}}$

Given:-

a1 = a2 = 1

Hence

Firstly we have to find

a1 , a2 , a3 , a4 , a5

For this we have to use one formula :-

$\tt{\rightarrow a_{n}=a_{n-1}+a_{n-2}}$

Here n > 2

According to information

$\tt{\rightarrow a_{1}=1}$

$\tt{\rightarrow a_{2}=1}$

Using Formula

When , n = 3

$\tt{\rightarrow a_{3}=a_{3-1}+a_{3-2}}$

$\tt{\rightarrow a_{3}=a_{2}+a_{1}}$

$\tt{\rightarrow a_{3}=a_{2}+a_{1}}$

Given,

$\tt{\rightarrow a_{2}=1\;and\;a_{1}=1}$

Hence,

= 1 + 1

= 2

When , n = 4

$\tt{\rightarrow a_{4}=a_{4-1}+a_{4-2}}$

$\tt{\rightarrow a_{4}=a_{3}+a_{2}}$

Given,

$\tt{\rightarrow a_{2}=1\;and\;a_{3}=2}$

= 2 + 1

= 3

When , n = 5

$\tt{\rightarrow a_{5}=a_{5-1}+a_{5-2}}$

$\tt{\rightarrow a_{5}=a_{4}+a_{3}}$

Given,

$\tt{\rightarrow a_{4}=3\;and\;a_{3}=2}$

= 3 + 2

= 5

When , n = 6

$\tt{\rightarrow a_{6}=a_{6-1}+a_{6-2}}$

$\tt{\rightarrow a_{6}=a_{5}+a_{4}}$

Given,

$\tt{\rightarrow a_{4}=3\;and\;a_{5}=5}$

= 5 + 3

= 8

Now,

Using Formula

$\tt{\rightarrow\dfrac{a_{n+1}}{a_{n}}}$

When, n = 1

$\tt{\rightarrow\dfrac{a_{1+1}}{a_{1}}}$

$\tt{\rightarrow\dfrac{a_{2}}{a_{1}}}$

$\tt{\rightarrow\dfrac{1}{1}}$

= 1

When, n = 2

$\tt{\rightarrow\dfrac{a_{2+1}}{a_{2}}}$

$\tt{\rightarrow\dfrac{a_{3}}{a_{2}}}$

$\tt{\rightarrow\dfrac{2}{1}}$

= 2

When, n = 3

$\tt{\rightarrow\dfrac{a_{3+1}}{a_{3}}}$

$\tt{\rightarrow\dfrac{a_{4}}{a_{3}}}$

$\tt{\rightarrow\dfrac{3}{2}}$

When, n = 4

$\tt{\rightarrow\dfrac{a_{4+1}}{a_{4}}}$

$\tt{\rightarrow\dfrac{a_{5}}{a_{4}}}$

$\tt{\rightarrow\dfrac{5}{3}}$

When, n = 5

$\tt{\rightarrow\dfrac{a_{5+1}}{a_{5}}}$

$\tt{\rightarrow\dfrac{a_{6}}{a_{5}}}$

$\tt{\rightarrow\dfrac{8}{5}}$