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### Definition of Homogeneous Recurrence Relation A homogeneous recurrence relation is a recurrence relation where each term in the sequence is defined solely in terms of previous terms, without any additional independent term. It can be expressed in the general form: \[ a_n + c_1 a_{n-1} + c_2 a_{n-2} + \dots + c_k a_{n-k} = 0 \] where \( c_1, c_2, \dots, c_k \) are constants. --- ### First-Order Homogeneous Recurrence Relation with Constant Coefficients A first-order recurrence relation depends only on the immediately preceding term and has the form: \[ a_n = c_1 a_{n-1} \] Example: \[ a_n = 3a_{n-1}, \quad \text{with } a_0 = 2 \] The general solution is \( a_n = 2 \cdot 3^n \). --- ### Second-Order Homogeneous Recurrence Relation with Constant Coefficients A ______ ______ ______ _______ _______ ___ _____ ____.
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