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Series whose k-th term is a ratio of standard sums

This type of question, asked 4 times: 27 Jul 2022, 8 Apr 2023, 2 Apr 2026 S2, 6 Apr 2026 S2. All 4 questions below, oldest first.

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  1. Question 1

    27 Jul 2022

    174
    23−131×7+43−33+23−132×11+\frac{2^{3}-1^{3}}{1 \times 7}+\frac{4^{3}-3^{3}+2^{3}-1^{3}}{2 \times 11}+ 63−53+43−33+23−133×15+…..+303−293+283−273+…+23−1315×63\begin{matrix} \frac{6^{3}-5^{3}+4^{3}-3^{3}+2^{3}-1^{3}}{3 \times 15}+ \ldots.. + \\ \frac{30^{3}-29^{3}+28^{3}-27^{3}+ \ldots +2^{3}-1^{3}}{15 \times 63} \end{matrix} is equal to

  2. Question 2

    8 Apr 2023

    73
    Let Sk=1+2+…..+KKS_{k}=\frac{1 + 2 + \ldots.. + K}{K} and ∑j=1nSj2=nA(Bn2+Cn+D)\sum_{j = 1}^{n} S_{j}^{2}=\frac{n}{A}\left( Bn^{2}+ Cn + D \right), where A,B,C,D∈NA,B,C,D \in N and AA has least value. Then

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  3. Question 3

    2 Apr 2026 S2

    55
    The sum 131+13+231+3+13+23+331+3+5+…\frac{1^{3}}{1}+\frac{1^{3}+2^{3}}{1 + 3}+\frac{1^{3}+2^{3}+3^{3}}{1 + 3 + 5}+ \ldots up to 8 terms, is :

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  4. Question 4

    6 Apr 2026 S2

    56
    The sum 1+12(12+22)+13(12+22+32)+...1 +\frac{1}{2}\left( 1^{2}+2^{2} \right)+\frac{1}{3}\left( 1^{2}+2^{2}+3^{2} \right)+ ... up to 10 terms is equal to :

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