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Infinite G.P. with a given sum and one more condition

This type of question, asked 3 times: 2021 Paper 18, 27 Jul 2022, 9 Apr 2024. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 18

    77
    If the sum of an infinite GP a,ar,ar2,ar3,…a,ar,ar^{2},ar^{3},\ldots is 15 and the sum of the squares of its each term is 150 , then the sum of ar2,ar4,ar6,…ar^{2},ar^{4},ar^{6},\ldots is:

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  2. Question 2

    27 Jul 2022

    157
    Let the sum of an infinite G.P., whose first term is aa and the common ratio is rr, be 5 . Let the sum of its first five terms be 9825\frac{98}{25}. Then the sum of the first 21 terms of an AP, whose first term is 10ar,nth 10ar,n^{\text{th~}} term is ana_{n} and the common difference is 10ar210ar^{2}, is equal to:

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  3. Question 3

    9 Apr 2024

    166
    Let a,ar,ar2a,ar,ar^{2}, ........be an infinite G.P. If ∑n=0∞arn=57\sum_{n= 0}^{\infty} ar^{n}= 57 and ∑n=0∞a3r3n=9747\sum_{n= 0}^{\infty} a^{3}r^{3n}= 9747, then a+18ra+ 18r is equal to :

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