For what value of n, an+1+bn+1an+bn is the arithmetic mean of a and b?
Since, arithmetic mean of a and b is a+b2, therefore according to the given condition,
an+1+bn+1an+bn=a+b2
⇒ 2an+1+2bn+1=an+1+anb+abn+bn+1
⇒ 2an+1−an+1−anb=abn+bn+1−2bn+1
⇒ an+1−anb=abn−bn+1
⇒ an(a−b)=bn(a−b)
⇒ an=bn [∵ a≠b]
⇒ (ab)n=1⇒(ab)n=(ab)0⇒ n=0
Hence, for n = 0, an+1+bn+1an+bn is the arithmetic mean of a and b.