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Sum of 1/(n(n+1)(n+2))

This type of question, asked 3 times: 29 Jul 2022, 29 Jan 2023, 5 Apr 2026 S1. All 3 questions below, oldest first.

Try each one first, then open its answer.

  1. Question 1

    29 Jul 2022

    88
    If 12×3×4+13×4×5+14×5×6+…+\frac{1}{2 \times 3 \times 4}+\frac{1}{3 \times 4 \times 5}+\frac{1}{4 \times 5 \times 6}+ \ldots + 1100×101×102=k101\frac{1}{100 \times 101 \times 102}=\frac{k}{101}, then 34k34k is equal to

  2. Question 2

    29 Jan 2023

    83
    Suppose ff is a function satisfying f(x+y)=f(x)+f(y)f(x + y) = f(x) + f(y) for all x,y∈Nx,y \in N and f(1)=15f(1) =\frac{1}{5}. If ∑n=1mf(n)n(n+1)(n+2)=112\sum_{n = 1}^{m} \frac{f(n)}{n(n + 1)(n + 2)}=\frac{1}{12}, then mm is equal to

  3. Question 3

    5 Apr 2026 S1

    55
    ∑n=110(528n(n+1)(n+2))\sum_{n = 1}^{10} \left( \frac{528}{n(n + 1)(n + 2)} \right) is equal to :

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