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Coefficients of x^4, x^5, x^6 in (1+x)^n are in A.P.: find n

The same question, asked 2 times: 4 Apr 2024*, 24 Jan 2025. Every wording below, oldest first.

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  1. Question 1

    4 Apr 2024

    163
    If the coefficients of x4,x5x^{4},x^{5} and x6x^{6} in the expansion of (1+x)n(1 + x)^{n} are in the arithmetic progression, then the maximum value of nn is:

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  2. Question 2

    24 Jan 2025

    66
    For some n≠10n \neq 10, let the coefficients of the 5th ,6th 5^{\text{th~}},6^{\text{th~}} and 7th 7^{\text{th~}} terms in the binomial expansion of (1+x)n+4(1 + x)^{n + 4} be in A.P. Then the largest coefficient in the expansion of (1+x)n+4(1 + x)^{n + 4} is :

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