SOLUTION: A sequence {{{a[n]}}} satisfies following condition. Calculate {{{lim( x->infinity, a[n] )}}} {{{a[1]=3}}}, {{{a[n+1]=(2/3)*a[n] + (1/4)}}} (n=1,2,3...)

Question 763244: A sequence satisfies following condition.
Calculate
, (n=1,2,3...)
Answer by DrBeeee(684) (Show Source): You can put this solution on YOUR website!
Let's assume that in the limit (a sub n) becomes a constant value c. then we have
(1) c = (2/3)*c + (1/4) or
(2) (1-2/3)*c = 1/4 or
(3) (1/3)*c = 1/4 or
(4) c = 3*(1/4) or
(5) c = 3/4 voila!!!
Check it
Is (3/4 = (2/3)*(3/4) + 1/4)?
Is (3/4 = 1/2 + 1/4)?
Is (3/4 = 3/4)? Yes
Answer: In the limit as n goes to infinity a sub n goes to 3/4.
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