(a ^ (n + 1) + b ^ (n + 1))/(a ^ n + b ^ n) is the G.M. between a and b. = sqrt(a) * sqrt(b) a^ prime prime +b^ prime prime sqrt a sqrt b Rightarrow sqrt(a) * a ^ n * (sqrt(a) – sqrt(b)) = b ^ n * sqrt(b) * (sqrt(a) – sqrt(b)) sqrt(a) * a ^ n = b ^ n * sqrt(b) *** sqrt(a) – sqrt b ne0 Ans. (a ^ (n + 1) + b ^ (n + 1))/(a ^ n + b ^ n) = sqrt(ab) Rightarrow a ^ n * sqrt(a) * sqrt(a) + b ^ n * sqrt(b) * sqrt(b) Rightarrow a ^ (n + 1) + b ^ (n + 1) = sqrt(a) * sqrt(b) * (a ^ n + b ^ n) (a/b) ^ n = (a/b) ^ (- 1/2) (a/b) ^ n = (sqrt(b/a)) n = – 1/2 = sqrt(a) * sqrt(b) b^ prime prime -b^ prime prime sqrt b * sqrt b Rightarrow a ^ n * sqrt(a) * sqrt(a) – sqrt(a) * sqrt(b) * a ^ n
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