Math Problem Statement

A sequence u1, u2, u3, ... is such that u1 = 1/4 and un+1 = un + (9 + n - 3n^2)/[(n+1)(n+2)]^2. Solve the sub-questions related to finding terms, proving a formula, summing a series, and evaluating an infinite sum.

Solution

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Math Problem Analysis

Mathematical Concepts

Sequences and Series
Mathematical Induction
Summation of Series
Infinite Series

Formulas

Recursive sequence relation: un+1 = un + (9 + n - 3n^2)/[(n+1)(n+2)]^2
Closed form for terms: un = (3n - 2)/(n+1)^2
Sum of series formula: sum from n=3 to infinity of f(n)

Theorems

Mathematical Induction

Suitable Grade Level

Grades 11-12 (Advanced High School or Early College Level)

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