anonymous
  • anonymous
Two large numbers of the Fibonacci sequence are F50 = 12,586,269,025 and F51 = 20,365,011,074. What is the approximate value of the quotient ? f51/50
Mathematics
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
A. 1.618 B. 0.618 C. 2.618 D. 2 E. 3.142
anonymous
  • anonymous
@Mehek14
anonymous
  • anonymous
we have that \(F_n\approx\frac1{\sqrt5}\varphi^n\) for large \(n\) where \(\varphi=\frac12(1+\sqrt5)\approx1.618\)

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anonymous
  • anonymous
so \(F_{n+1}/F_n\approx \varphi\approx 1.618\) for large \(n\); in this case, \(n=50\) is sufficiently big for this approximation to be 'good'
anonymous
  • anonymous
so it =1.618
anonymous
  • anonymous
????? @oldrin.bataku
anonymous
  • anonymous
check for yourself if you have a calculator; even just 3 digits of each are enough to give a decent approximation
anonymous
  • anonymous
\(F_{51}/F_{50}\approx 204/126=1.619\approx1.618\)

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