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