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Infinite series whose numerators have constant second differences over powers of a number

This type of question, asked 4 times: 2021 Paper 23, 27 Jun 2022, 29 Jun 2022, 11 Apr 2023. All 4 questions below, oldest first.

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  1. Question 1

    2021 Paper 23

    86
    If S=75+952+1353+1954+…S=\frac{7}{5}+\frac{9}{5^{2}}+\frac{13}{5^{3}}+\frac{19}{5^{4}}+ \ldots, then 160 S160\text{ }S is equal to

  2. Question 2

    27 Jun 2022

    155
    Let S=2+67+1272+2073+3074+…S = 2 +\frac{6}{7}+\frac{12}{7^{2}}+\frac{20}{7^{3}}+\frac{30}{7^{4}}+ \ldots. then 4S4S is equal to

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  3. Question 3

    29 Jun 2022

    154
    The sum of the infinite series 1+56+1262+2263+3564+5165+7066+…1 +\frac{5}{6}+\frac{12}{6^{2}}+\frac{22}{6^{3}}+\frac{35}{6^{4}}+\frac{51}{6^{5}}+\frac{70}{6^{6}}+ \ldots. is equal to:

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  4. Question 4

    11 Apr 2023

    175
    For k∈Nk \in N, if the sum of the series 1+4k+8k2+13k3+19k4+…1 +\frac{4}{k}+\frac{8}{k^{2}}+\frac{13}{k^{3}}+\frac{19}{k^{4}}+ \ldots is 10, then the value of kk is