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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+1an+1anb=abn+bn+12bn+1

an+1anb=abnbn+1

an(ab)=bn(ab)

an=bn [ ab]

(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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