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Sum of an arithmetico-geometric series (A.P. terms times G.P. terms)

This type of question, asked 9 times: 2021 Paper 14, 2021 Paper 25, 25 Jun 2022, 29 Jul 2022, 6 Apr 2023, 10 Apr 2023, 11 Apr 2023, 27 Jan 2024, 24 Jan 2025. All 9 questions below, oldest first.

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  1. Question 1

    2021 Paper 14

    57
    If (1+23+632+1033…∞)log⁡(0.25)(13+132+133…∞)=l\left( 1 +\frac{2}{3}+\frac{6}{3^{2}}+\frac{10}{3^{3}}\ldots\infty \right)^{\log_{\left( 0.25 \right)}\left( \frac{1}{3}+\frac{1}{3^{2}}+\frac{1}{3^{3}}\ldots\infty \right)}=l then find l2l^{2}

  2. Question 2

    2021 Paper 25

    68
    The sum of the infinite series 1+23+732+1233+1734+2235+…1 +\frac{2}{3}+\frac{7}{3^{2}}+\frac{12}{3^{3}}+\frac{17}{3^{4}}+\frac{22}{3^{5}}+ \ldots is equal to

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  3. Question 3

    25 Jun 2022

    158
    The sum 1+2⋅3+3⋅32+…..+10⋅391 + 2 \cdot 3 + 3 \cdot3^{2}+ \ldots.. + 10 \cdot3^{9} is equal to

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  4. Question 4

    29 Jul 2022

    84
    Let a1,a2,a3,…a_{1},a_{2},a_{3},\ldots be an A.P. If ∑r=1∞ar2r=4\sum_{r = 1}^{\infty} \frac{a_{r}}{2^{r}}= 4, then 4a24a_{2} is equal to

  5. Question 5

    6 Apr 2023

    174
    If (20)19+2(21)(20)18+3(21)2(20)17+(20)^{19}+ 2(21)(20)^{18}+ 3(21)^{2}(20)^{17}+ +20(21)19=k(20)19+ 20(21)^{19}= k(20)^{19}, then kk is equal to

  6. Question 6

    10 Apr 2023

    176
    Suppose a1,a2,2,a3,a4a_{1},a_{2},2,a_{3},a_{4} be in an arithmeticogeometric progression. If the common ratio of the corresponding geometric progression is 2 and the sum of all 5 terms of the arithmetico-geometric progression is 492\frac{49}{2}, then a4a_{4} is equal to

  7. Question 7

    11 Apr 2023

    82
    Let S=109+1085+10752+……..+25107+15108S = 109 +\frac{108}{5}+\frac{107}{5^{2}}+ \ldots\ldots.. +\frac{2}{5^{107}}+\frac{1}{5^{108}}. Then the value of (16 S−(25)−54)\left( 16\text{ }S - (25)^{- 54} \right) is equal to

  8. Question 8

    27 Jan 2024

    85
    If 8=3+14(3+p)+142(3+2p)+143(3+3p)+…∞8 = 3 +\frac{1}{4}(3 + p) +\frac{1}{4^{2}}(3 + 2p) +\frac{1}{4^{3}}(3 + 3p) + \ldots\infty then the value of pp is

  9. Question 9

    24 Jan 2025

    135
    If 7=5+17(5+α)+172(5+2α)+173(5+3α)+⋯7 = 5 + \frac{1}{7}(5+\alpha) + \frac{1}{7^{2}}(5+2\alpha) + \frac{1}{7^{3}}(5+3\alpha) + \cdots upto ∞\infty, then the value of α\alpha is:

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