We know the arithmetic mean of ‘a’ and ‘b’ is (a+b)/2
Hence (a+b)/2 = (an+1+bn+1)/(an+bn)
(a+b)/2 = (an x a + bn x b)/(an+bn)
(ana + anb + bna + bnb) = (2ana+2bnb)
(anb + bna) = (ana + bnb)
an(b – a)=bn(b – a)
an = bn
This is possible only when n = 0