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Numbers in A.P. (or G.P.) become a G.P. (or A.P.) after given changes

This type of question, asked 5 times: 2021 Paper 22, 1 Feb 2024, 4 Apr 2024, 7 Apr 2025, 22 Jan 2026 S2. All 5 questions below, oldest first.

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  1. Question 1

    2021 Paper 22

    68
    Three numbers are in an increasing geometric progression with common ratio rr. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference dd. If the fourth term of GP is 3r23r^{2}, then r2−dr^{2}-d is equal to:

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  2. Question 2

    1 Feb 2024

    77
    Let 3,a,b,c3,a,b,c be in A.P. and 3,a−1,b+1,c+93,a- 1,b+ 1,c+ 9 be in G.P. Then, the arithmetic mean of a,ba,b and cc is:

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  3. Question 3

    4 Apr 2024

    155
    Let three real numbers a,b,ca,b,c be in arithmetic progression and a+1, b,c+3a + 1,\text{ }b,c + 3 be in geometric progression. If a>10a > 10 and the arithmetic mean of a,ba,b and cc is 8 , then the cube of the geometric mean of a,ba,b and cc is

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  4. Question 4

    7 Apr 2025

    63
    Let x1,x2,x3,x4x_{1},x_{2},x_{3},x_{4} be in a geometric progression. If 2,7,9,52,7,9,5 are subtracted respectively from x1,x2,x3x_{1},x_{2},x_{3}, x4x_{4} then the resulting numbers are in an arithmetic progression. Then the value of 124(x1x2x3x4)\frac{1}{24}\left( x_{1}x_{2}x_{3}x_{4} \right) is :

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  5. Question 5

    22 Jan 2026 S2

    74
    Suppose a,b,ca,b,c are in A.P. and a2,2 b2,c2a^{2},2{\text{ }b}^{2},c^{2} are in G.P. If a<b<ca < b < c and a+b+c=1a + b + c = 1, then 9(a2+b2+c2)9\left( a^{2} + b^{2} + c^{2} \right) is equal to ____\_\_\_\_.