Let f(n) = (4n + sqrt(4n² - 1)) / (sqrt(2n+1) + sqrt(2n-1)) for n in N. If r is the remainder when S = f(1) + f(2) +... + f(60) is divided by 9, which of the following is/are correct?

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r = 8. After rationalizing: f(n) simplifies to sqrt(2n+1)*sqrt(2n-1)... let's try: numerator = 4n + sqrt(2n+1)*sqrt(2n-1), denominator = sqrt(2n+1)+sqrt(2n-1). Multiply top and bottom by (sqrt(2n+1)-sqrt(2n-1)): denom becomes (2n+1)-(2n-1)=2. Numerator: [4n+sqrt((2n+1)(2n-1))]*(sqrt(2n+1)-sqrt(2n-1)) = 4n*sqrt(2n+1)-4n*sqrt(2n-1)+(2n+1)*sqrt(2n-1)... this gets complex. Alternatively write f(n) = (sqrt(2n+1)+sqrt(2n-1))²/2 / (sqrt(2n+1)+sqrt(2n-1))/... Simpler: note (sqrt(2n+1)+sqrt(2n-1))*(sqrt(2n+1)-sqrt(2n-1))=2, and 4n=(2n+1)+(2n-1)=(sqrt(2n+1))²+(sqrt(2n-1))². So f(n) = [(sqrt(2n+1))²+(sqrt(2n-1))²+sqrt(2n+1)*sqrt(2n-1)]/(sqrt(2n+1)+sqrt(2n-1)). This equals (sqrt(2n+1))²+... hmm. Try dir

Let f(n) = (4n + sqrt(4n² - 1)) / (sqrt(2n+1) + sqrt(2n-1)) for n in N. If r is the remainder when S = f(1) + f(2) +... + f(60) is divided by 9, which of the following is/are correct?

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