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)) }
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