If the first three Fibonacci nos. are x1=1, x2=1, x3=2, what is the least value of n for which xn+1 /n=1.619 correct to the three decimal places ?

n-th Fibonacci numbers f(n)={[(5^0.5+1)/2]^n - [(5^0.5–1)/2]^n}/(5^0.5)

(Bine’s formule)

[(5^0.5–1)/2]<1 therefore [(5^0.5–1)/2]^n is near zero when n is large.

So

f(n)={[(5^0.5+1)/2]^n}/(5^0.5)>5000

ln{f(n)}=n*ln{[(5^0.5+1)/2]} > ln[5000*(5^0.5)]

n*ln{[(5^0.5+1)/2]}> ln[5000*(5^0.5)]

n > ln[5000*(0.5*ln5)]/ln{[(5^0.5+1)/2]} = 19,37174

So f(19)<5000, f(20)>5000

Step-by-step explanation:

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step-by-step explanation:

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