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...)

Algebra ->  Sequences-and-series -> 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...)      Log On


   

Question 763244: A sequence a%5Bn%5D satisfies following condition.
Calculate lim%28+x-%3Einfinity%2C+a%5Bn%5D+%29
a%5B1%5D=3, a%5Bn%2B1%5D=%282%2F3%29%2Aa%5Bn%5D+%2B+%281%2F4%29 (n=1,2,3...)
Answer by DrBeeee(684) About Me  (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.