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Ratio or relation of power sums of the roots, simplified with the recurrence from the quadratic

This type of question, asked 8 times: 2021 Paper 4, 2021 Paper 26, 29 Jul 2022, 27 Jan 2024, 6 Apr 2024, 9 Apr 2024, 2 Apr 2025, 3 Apr 2025. All 9 questions below, oldest first.

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  1. Question 1

    2021 Paper 4

    75
    Let α\alpha and β\beta be the roots of x2−6x−2=0x^{2}- 6x- 2 = 0. If an=αn−βna_{n}=\alpha^{n}-\beta^{n} for n≥1n\geq 1, then the value of a10−2a83a9\frac{a_{10}- 2a_{8}}{3a_{9}} is:

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  2. Question 2

    2021 Paper 26

    82
    Let α\alpha and β\beta be two real numbers such that α+β=1\alpha + \beta = 1 and αβ=−1\alpha\beta = - 1. Let Pn=(α)n+(β)n,Pn−1=11P_{n}= (\alpha)^{n}+ (\beta)^{n},P_{n - 1}= 11 and Pn+1=29P_{n + 1}= 29 for some integer n≥1n \geq 1. Then, the value of Pn2P_{n}^{2} is

  3. Question 3

    29 Jul 2022

    172
    Let α,β(α>β)\alpha,\beta(\alpha > \beta) be the roots of the quadratic equation x2−x−4=0x^{2}- x - 4 = 0. If Pn=αn−βn,n∈NP_{n}=\alpha^{n}-\beta^{n},n \in N, then P15P16−P14P16−P152+P14P15P13P14\frac{P_{15}P_{16}-P_{14}P_{16}-P_{15}^{2}+P_{14}P_{15}}{P_{13}P_{14}} is equal to

  4. Question 4

    27 Jan 2024

    160
    If α,β\alpha,\beta are the roots of the equation, x2−x−1=0x^{2}- x - 1 = 0 and Sn=2023αn+2024βnS_{n}= 2023\alpha^{n}+ 2024\beta^{n}, then

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  5. Question 5

    6 Apr 2024

    178
    Let α,β\alpha,\beta be roots of x2+2x−8=0x^{2}+\sqrt{2}x - 8 = 0. If Un=αn+βnU_{n}=\alpha^{n}+\beta^{n}, then U10+2U92U8\frac{U_{10}+\sqrt{2}U_{9}}{2U_{8}} is equal to

  6. Question 6

    6 Apr 2024

    71
    Let, α,β\alpha,\beta be the distinct roots of the equation x2−(t2−5t+6)x+1=0,t∈Rx^{2}-\left( t^{2}- 5t + 6 \right)x + 1 = 0,t \in R and an=αn+βna_{n}=\alpha^{n}+\beta^{n}. Then the minimum value of a2023+a2025a2024\frac{a_{2023}+a_{2025}}{a_{2024}} is

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  7. Question 7

    9 Apr 2024

    169
    Let α,β;α>β\alpha,\beta;\alpha > \beta, be the roots of the equation x2−2x−3=0x^{2}-\sqrt{2}x -\sqrt{3}= 0. Let Pn=αn−βn,n∈NP_{n}=\alpha^{n}-\beta^{n},n \in N. Then (113−102)P10+(112+10)P11−11P12(11\sqrt{3}- 10\sqrt{2})P_{10}+ (11\sqrt{2}+ 10)P_{11}- 11P_{12} is equal to:

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  8. Question 8

    2 Apr 2025

    65
    Let Pn=αn+βn,n∈NP_{n}=\alpha^{n}+\beta^{n},n\in N. If P10=123,P9=76P_{10}= 123,P_{9}= 76, P8=47P_{8}= 47 and P1=1P_{1}= 1, then the quadratic equation having roots 1α\frac{1}{\alpha} and 1β\frac{1}{\beta} is:

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  9. Question 9

    3 Apr 2025

    53
    Let α\alpha and β\beta be the roots of x2+3x−16=0x^{2}+\sqrt{3}x- 16 = 0, and γ\gamma and δ\delta be the roots of x2+3x−1=0x^{2}+ 3x - 1 = 0. If Pn=αn+βnP_{n}=\alpha^{n}+\beta^{n} and Qn=γn+δnQ_{n}=\gamma^{n}+\delta^{n}, then P25+3P242P23+Q25−Q23Q24\frac{P_{25}+\sqrt{3}P_{24}}{2P_{23}}+\frac{Q_{25}-Q_{23}}{Q_{24}} is equal to

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