F_n = F_n _(−1) +F_(n−2) F_2 = F_1 =1 F_n : 1 , 1 , 2 , 3 ,5... f(x)= Σ_(n=1) ^∞ F_n x^( n) = x + x^( 2) +Σ_(n=3) ^∞ (F_(n−1) +F_(n−2) )x^( n) = x+x^2 + Σ_(n=3) ^∞ F_(n−1) x^( n) + x^( 2) f (x) = x + x^( 2) + x^( 2) f(x) +x Σ_(n=2) ^∞ F_n x^( n) = x + x^( 2) + x^( 2) f(x)−x^( 2) + xf(x) ∴ f(x)= (x/(1−x−x^( 2) )) (generating function ) (x/(1−x−x^( 2) )) =Σ_(n=1) ^∞ F_n x^( n) ⇒ (x^( 2) /(1−x−x^( 2) ))=Σ_(n=1) ^∞ F_n x^( n+1) x= (1/(10)) ⇒ ((1/(100))/(1−(1/(10))−(1/(100)))) = Σ_(n=1) ^∞ (F_n /(10^( n+1) )) ⇒ { Σ_(n=1) ^∞ (( F


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Question Number 194020 by mnjuly1970 last updated on 26/Jun/23
$F_n = F_n _(−1) +F_(n−2) F_2 = F_1 =1 F_n : 1 , 1 , 2 , 3 ,5... f(x)= Σ_(n=1) ^∞ F_n x^( n) = x + x^( 2) +Σ_(n=3) ^∞ (F_(n−1) +F_(n−2) )x^( n) = x+x^2 + Σ_(n=3) ^∞ F_(n−1) x^( n) + x^( 2) f (x) = x + x^( 2) + x^( 2) f(x) +x Σ_(n=2) ^∞ F_n x^( n) = x + x^( 2) + x^( 2) f(x)−x^( 2) + xf(x) ∴ f(x)= (x/(1−x−x^( 2) )) (generating function ) (x/(1−x−x^( 2) )) =Σ_(n=1) ^∞ F_n x^( n) ⇒ (x^( 2) /(1−x−x^( 2) ))=Σ_(n=1) ^∞ F_n x^( n+1) x= (1/(10)) ⇒ ((1/(100))/(1−(1/(10))−(1/(100)))) = Σ_(n=1) ^∞ (F_n /(10^( n+1) )) ⇒ { Σ_(n=1) ^∞ (( F_n )/(10^( n+1) )) = (1/(89)) }$
$F_{n} = F_{n}_{- 1} + F_{n - 2} F_{2} = F_{1} = 1$ $F_{n} : 1, 1, 2, 3, 5 . . .$ $f (x) = \underset{n = 1}{\sum^{\infty}} F_{n} x^{n} = x + x^{2} + \underset{n = 3}{\sum^{\infty}} (F_{n - 1} + F_{n - 2}) x^{n}$ $= x + x^{2} + \underset{n = 3}{\sum^{\infty}} F_{n - 1} x^{n} + x^{2} f (x)$ $= x + x^{2} + x^{2} f (x) + x \underset{n = 2}{\sum^{\infty}} F_{n} x^{n}$ $= x + x^{2} + x^{2} f (x) - x^{2} + x f (x)$ $∴ f (x) = \frac{x}{1 - x - x^{2}} (g e n e r a t i n g f u n c t i o n)$ $\frac{x}{1 - x - x^{2}} = \underset{n = 1}{\sum^{\infty}} F_{n} x^{n} \Rightarrow \frac{x^{2}}{1 - x - x^{2}} = \underset{n = 1}{\sum^{\infty}} F_{n} x^{n + 1}$ $x = \frac{1}{10} \Rightarrow \frac{\frac{1}{100}}{1 - \frac{1}{10} - \frac{1}{100}} = \underset{n = 1}{\sum^{\infty}} \frac{F_{n}}{10^{n + 1}}$ $\Rightarrow {\underset{n = 1}{\sum^{\infty}} \frac{F_{n}}{10^{n + 1}} = \frac{1}{89}}$
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