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To solve the recurrence relation using the generating function technique, follow these steps: ### Step 1: Define the Generating Function The given recurrence relation is: \[ a_n = 2a_{n-1} + 1, \quad \text{for } n \geq 1, \quad \text{with } a_0 = 0. \] Define the generating function as: \[ A(x) = \sum_{n=0}^{\infty} a_n x^n. \] ### Step 2: Form the Generating Function Equation Multiply both sides of the recurrence by \( x^n \) and sum over all \( n \): \[ \sum_{n=1}^{\infty} a_n x^n = \sum_{n=1}^{\infty} (2a_{n-1} + 1) x^n. \] Splitting the sum, \[ \sum_{n=1}^{\infty} a_n x^n = 2 \sum_{n=1}^{\infty} a_{n-1} x^n + \sum_{n=1}^{\infty} ___ _________ ____ ______ __________ ________ __________ ______ ____ ________.
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