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We are given the sum: $$ S_n = 1 + 2 + 3 + \dots + n $$ We are to derive this using generating functions, not just quote the formula. --- ### ✅ Step 1: Consider the sequence We define the sequence: $$ a_n = n \quad \text{for } n \geq 1,\quad a_0 = 0 $$ So the sequence is: $$ a_0 = 0,\ a_1 = 1,\ a_2 = 2,\ a_3 = 3,\ \dots $$ Let the generating function be: $$ A(x) = \sum_{n=0}^\infty a_n x^n = \sum_{n=1}^\infty __________ ____ ______ ______ _______ _____ ____ _______ ________.
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