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Infinite sum of brackets (a^(n+1) - b^(n+1))/(a - b)-type groups of two geometric ratios

The same question, asked 2 times: 15 Apr 2023*, 24 Jan 2026 S2. Every wording below, oldest first.

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  1. Question 1

    15 Apr 2023

    81
    If the sum of the series
    (12−13)+(122−12⋅3+132)+(123−122⋅3+12⋅32−133)+(124−123⋅3+122⋅32−12⋅33+134)+…. is αβ, where \begin{matrix} & \left( \frac{1}{2}-\frac{1}{3} \right)+\left( \frac{1}{2^{2}}-\frac{1}{2 \cdot 3}+\frac{1}{3^{2}} \right)+ \\ & \left( \frac{1}{2^{3}}-\frac{1}{2^{2}\cdot 3}+\frac{1}{2 \cdot3^{2}}-\frac{1}{3^{3}} \right)+ \\ & \left( \frac{1}{2^{4}}-\frac{1}{2^{3}\cdot 3}+\frac{1}{2^{2}\cdot3^{2}}-\frac{1}{2 \cdot3^{3}}+\frac{1}{3^{4}} \right)+ \ldots.\text{~is~}\frac{\alpha}{\beta},\text{~where~} \end{matrix}
    α\alpha and β\beta are co-prime, then α+3β\alpha + 3\beta is equal to....

  2. Question 2

    24 Jan 2026 S2

    57
    (13+47)+(132+13×47+4272)+\left( \frac{1}{3}+\frac{4}{7} \right)+\left( \frac{1}{3^{2}}+\frac{1}{3}\times\frac{4}{7}+\frac{4^{2}}{7^{2}} \right)+ (133+132×47+13×4272+4373)+……\left( \frac{1}{3^{3}}+\frac{1}{3^{2}}\times\frac{4}{7}+\frac{1}{3}\times\frac{4^{2}}{7^{2}}+\frac{4^{3}}{7^{3}} \right)+ \ldots\ldots upto infinite terms is equal to -

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