JEEOlympiad
The sum S = sum_{n=1}^{80} 1/((n+1)*sqrt(n) + n*sqrt(n+1)) equals m/n…
Question
The sum S = sum_{n=1}^{80} 1/((n+1)*sqrt(n) + n*sqrt(n+1)) equals m/n in lowest terms, where m and n are coprime positive integers. Find m + n.
✓ Verified answer: 17checked by our engine — not a guess
Step-by-step solution
Factor the denominator: (n+1)sqrt(n) + n sqrt(n+1) = sqrt(n)*sqrt(n+1)*(sqrt(n+1) + sqrt(n)).
So the term = 1/(sqrt(n)sqrt(n+1)(sqrt(n+1)+sqrt(n))). Rationalize by multiplying numerator and denominator by (sqrt(n+1) - sqrt(n)):
term = (sqrt(n+1) - sqrt(n))/(sqrt(n)sqrt(n+1)) = 1/sqrt(n) - 1/sqrt(n+1).
This telescopes: S = 1/sqrt(1) - 1/sqrt(81) = 1 - 1/9 = 8/9.
m/n = 8/9, so m + n = 17.
Final answer17
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