\frac{1}{k(k+1)(k+2)} = \frac{A}{k} + \frac{B}{k+1} + \frac{C}{k+2}.
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- Final answer: $\boxed{(4, 3)}$
- Question: Microfossil density $ d(t) $ in a sediment layer follows $ \sum_{k=1}^{n} \frac{1}{k(k+1)(k+2)} $. Compute this sum for $ n = 50 $.
- Solution: Use partial fractions to decompose $ \frac{1}{k(k+1)(k+2)} $. Let:
- Multiply through: $ 1 = A(k+1)(k+2) + Bk(k+2) + Ck(k+1) $.
- Solve for $ A, B, C $:
- For $ k = 0 $: $ 1 = A(1)(2) \implies A = \frac{1}{2} $.
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