Math Problem Statement

1. Solve t_n - 3t_{n-1} = 4n(2n+1) for n ≥ 0 with t_0 = 12. 2. Solve a_n = 9a_{n-1} - 15a_{n-2} + 7a_{n-3} with given conditions a_0 = 0, a_1 = 1, a_2 = 2. 3. Determine the generating function for P(x) = x^5 / (1 - 8x) and P(x) = 1/2 (x^2 + e^(-x)). 4. Solve the recurrence relation a_n = 3a_{n-1} + 2n for a_1 = 3.

Solution

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Here are the questions:

  1. Solve tn−3tn−1=4n(2n+1)t_n - 3t_{n-1} = 4n(2n+1).
  2. Solve an=9an−1−15an−2+7an−3a_n = 9a_{n-1} - 15a_{n-2} + 7a_{n-3}, with given conditions.
  3. Find generating functions P(x)P(x).
  4. Solve an=3an−1+2na_n = 3a_{n-1} + 2n, a non-homogeneous recurrence relation.

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Math Problem Analysis

Mathematical Concepts

Recurrence Relations
Non-Homogeneous Linear Recurrence
Generating Functions
Linear Algebra

Formulas

General solution for recurrence relation: a_n = C * r^n + particular solution.
Generating functions: P(x) = Σ a_n * x^n.
Non-homogeneous solution: a_n = A * r^n + particular terms.

Theorems

Linear Recurrence Theorem
Generating Function Method
Superposition Principle for Non-Homogeneous Recurrence Relations

Suitable Grade Level

Undergraduate or Advanced High School (Grade 12+)

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