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Ratio of the k-th term from the beginning to the k-th term from the end

This type of question, asked 4 times: 29 Jul 2022, 6 Apr 2023, 11 Apr 2023, 4 Apr 2025. All 4 questions below, oldest first.

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  1. Question 1

    29 Jul 2022

    85
    Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of (24+134)n\left( \sqrt[4]{2}+\frac{1}{\sqrt[4]{3}} \right)^{n}, in the increasing powers of 134\frac{1}{\sqrt[4]{3}} be 64:1\sqrt[4]{6}:1. If the sixth term from the beginning is α34\frac{\alpha}{\sqrt[4]{3}}, then α\alpha is equal to

  2. Question 2

    6 Apr 2023

    64
    If the ratio of the fifth term from the begining to the fifth term from the end in the expansion of (24+134)n\left( \sqrt[4]{2}+\frac{1}{\sqrt[4]{3}} \right)^{n} is 6:1\sqrt{6}:1, then the third term from the beginning is:

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  3. Question 3

    11 Apr 2023

    153
    If the 1011th 1011^{\text{th~}} term from the end in the binomial expansion of (4x5−52x)2022\left( \frac{4x}{5}-\frac{5}{2x} \right)^{2022} is 1024 times 1011th 1011^{\text{th~}} term from the beginning, then ∣x∣|x| is equal to

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  4. Question 4

    4 Apr 2025

    62
    In the expansion of (23+133)n,n∈N\left( \sqrt[3]{2}+\frac{1}{\sqrt[3]{3}} \right)^{n},n \in N, if the ratio of 15th 15^{\text{th~}} term from the beginning to the 15th 15^{\text{th~}} term from the end is 16\frac{1}{6}, then the value of  nC3\ ^{n}C_{3} is:

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