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  • The Fibonacci sequence is defined by 1 = a1 = a2 and an = an – 1 + an – 2, n > 2. Find n 1 n a a + , for n = 1, 2, 3, 4, 5

14.   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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best_answer

Given : The Fibonacci sequence is defined by 1 = a _ 1 = a _2 \: \:and \: \: a _n = a _{n-1} + a _{n-2} , 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=5

a _6 = a _{6-1} + a _{6-2} =a_5+a_4=5+3=8

 For \,\,n=1,\frac{a _{n+1}}{a_n}=\frac {a_{1+1}}{a_1}=\frac{a_2}{a_1}=\frac{1}{1}=1

For \,\, n=2,\frac{a _{n+1}}{a_n}=\frac {a_{2+1}}{a_2}=\frac{a_3}{a_2}=\frac{2}{1}=2

For \,\, n=3,\frac{a _{n+1}}{a_n}=\frac {a_{3+1}}{a_3}=\frac{a_4}{a_3}=\frac{3}{2}

For \,\, n=4,\frac{a _{n+1}}{a_n}=\frac {a_{4+1}}{a_4}=\frac{a_5}{a_4}=\frac{5}{3}

For \,\, n=5,\frac{a _{n+1}}{a_n}=\frac {a_{5+1}}{a_5}=\frac{a_6}{a_5}=\frac{8}{5}

 

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seema garhwal

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