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Three numbers in A.P. become a G.P. when given numbers are added: find their mean

The same question, asked 2 times: 1 Feb 2024*, 4 Apr 2024. Every wording below, oldest first.

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  1. Question 1

    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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  2. Question 2

    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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