Person A offers something for sale at a1=£a and person B offers b1=£b.

If A splits the difference and both continue indefinitely, where would it end?

Answer is £(a+2b)/3 ie nearer to £b than £a

Solution:

a2= (a+b)/2        b2=(a2+b)/2 =(a+3b)/4      a3=(a2+b2)/4=(3a+5b)/8

Similarly b3=(5a+11b)/16     a4=(11a+21b)/32     b4=(21a+43b)/64  etc

Two successive terms take the  form     au          + b(1-u) 

                                              then     a(1-u)/2 + b(1+u)/2

Assuming the convergence of (ratio of) coefficients of a and b:

(a)... u/(1-u) = (1-u)/(1+u) ... (b)    which yields u=1/3   QED

 

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