Whats the Ratio?

Calculus Level 3

Suppose the sequence where b ≥ a > 0 b\ge a> 0

a , b , a + b , a + 2 b , 2 a + 3 b , 3 a + 5 b , 5 a + 8 b , … a, b, a+b, a+2b, 2a +3b, 3a+5b, 5a+8b, \ldots

which follows the pattern t n = t n − 1 + t n − 2 { t }_{ n }={ t }_{ n-1 }+{ t }_{ n-2 }

Evaluate lim ⁡ n → ∞ t n + 1 t n . \lim \limits_{ n\to \infty }{ \frac { { t }_{ n+1 } }{ { t }_{ n } } } .


The answer is 1.61803398875.

Will McGlaughlin by Will McGlaughlin , 21, Australia

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1 solution

Will McGlaughlin
May 9, 2018

All sequences that follow this additive property t n = t n − 1 + t n − 2 { t }_{ n }={ t }_{ n-1 }+{ t }_{ n-2 } will approach the golden ratio when the ratio is taken from t n + 1 t n \frac{{ t }_{ n +1 }}{{ t }_{ n }}

Therefore lim ⁡ n → ∞ t n + 1 t n = Φ \lim _{ n\rightarrow \infty }{ \frac { { t }_{ n+1 } }{ { t }_{ n } } } =\boxed{\Phi }

1 Helpful 0 Interesting 0 Brilliant 1 Confused

I don't understand why it must be Φ \Phi and not 1 − Φ 1-\Phi ?

Joe Mansley - 3 months, 2 weeks ago

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