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Sequence defined by a linear recurrence

This type of question, asked 3 times: 29 Jun 2022, 29 Jul 2022, 28 Jan 2025. All 3 questions below, oldest first.

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  1. Question 1

    29 Jun 2022

    72
    Let {an}n=0∞\left\{ a_{n} \right\}_{n = 0}^{\infty} be a sequence such that a0=a1=0a_{0}=a_{1}= 0 and an+2=2an+1−an+1a_{n + 2}= 2a_{n + 1}-a_{n}+ 1 for all n≥0n \geq 0. Then, ∑n=2∞an7n\sum_{n = 2}^{\infty} \frac{a_{n}}{7^{n}} is equal to

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  2. Question 2

    29 Jul 2022

    156
    Let {an}n=0∞\left\{ a_{n} \right\}_{n = 0}^{\infty} be a sequence such that a0=a1=0a_{0}=a_{1}= 0 and an+2=3an+1−2an+1,∀n≥0a_{n + 2}= 3a_{n + 1}- 2a_{n}+ 1,\forall n \geq 0. Then a25a23−2a25a22−2a23a24+4a22a24a_{25}a_{23}- 2a_{25}a_{22}- 2a_{23}a_{24}+ 4a_{22}a_{24} is equal to:

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  3. Question 3

    28 Jan 2025

    59
    Let ⟨an⟩\left\langle a_{n} \right\rangle be a sequence such that a0=0,a1=12a_{0}= 0,a_{1}=\frac{1}{2} and 2an+2=5an+1−3an,n=0,1,2,3,……2a_{n + 2}= 5a_{n + 1}- 3a_{n},n = 0,1,2,3,\ldots\ldots Then ∑k=1100ak\sum_{k = 1}^{100} a_{k} is equal to :

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