Sum of an infinite series whose numerators have constant second differences over powers of a number

The same question, asked 3 times: 2021 Paper 23*, 27 Jun 2022*, 29 Jun 2022. Every wording below, oldest first.

Try each one first, then open its answer.

Question 1
2021 Paper 23
86 $S=\frac{7}{5}+\frac{9}{5^{2}}+\frac{13}{5^{3}}+\frac{19}{5^{4}}+ \ldots$
Question 2
27 Jun 2022
155 $S = 2 +\frac{6}{7}+\frac{12}{7^{2}}+\frac{20}{7^{3}}+\frac{30}{7^{4}}+ \ldots$
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Question 3
29 Jun 2022
154 $1 +\frac{5}{6}+\frac{12}{6^{2}}+\frac{22}{6^{3}}+\frac{35}{6^{4}}+\frac{51}{6^{5}}+\frac{70}{6^{6}}+ \ldots$
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