In the infinite sequence
a, b, (a + b)/2, (a + 3b)/4, . . .
each term after the second is the arithmetic mean of the two previous terms.

Find the limit of the sequence in terms of real numbers a and b.

Looking at the series  ...1 1 3 5 11 21 43  85 (each 2 consecutive numbers M  and  N   correspond to coefficients of a and b  in the expression ( Ma+Nb)/2^k  )  one gets the explicit formula of

M=(2^n +(-1)^n )/3  and         N=(2^(n +1)+(-1)^(n +1))/3

If n  is large enough the term ( -1)^n is ignored and we get

(2^n*a+2^(n+1))/(3*2^n)=  ( a+2b)/3

Posted by Ady TZIDON on 2007-05-23 20:22:25