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The sum S = [1*2² + 2*3² + 3*4² +... + n*(n+1)²] / [1²*2 + 2²*3 + 3²*4 +... + n²*(n+1)] is expressed in the form (3n + b)/(3n + c), where 3 is a prime number. Find the value of b + c (note: a = 3 is the prime).

1. 9
2. 7
3. 6
4. 4

Correct answer: 7

Solution

Both the numerator and denominator are polynomial in n of degree 4; dividing and simplifying yields a ratio of the form (3n+5)/(3n+2) where a=3 (prime), b=5, c=2, giving b+c=7.

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