Question

# For what value of n, an+1+bn+1an+bn is the arithmetic mean of a and b?

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Solution

## 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.

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