IGNOU BMTC 103 2026 - Find the generating function of the recurrence with initial

Question

Find the generating function of the recurrence $a_n = 6a_{n-1} - 5a_{n-2} + 1$ with initial conditions $a_0 = 2, a_1 = 5$ .

20 Jan 2026

Answer :

Word Count : 269

Let (A(x) = \sum_{n=0}^{\infty} a_n x^n) be the generating function of the sequence ({a_n}). The recurrence is [ a_n = 6a_{n-1} - 5a_{n-2} + 1, \quad a_0 = 2, , a_1 = 5. ] Multiply both sides of the recurrence by (x^n) and sum over (n \ge 2): [ \sum_{n=2}^{\infty} a_n x^n = \sum_{n=2}^{\infty} 6 a_{n-1} x^n - \sum_{n=2}^{\infty} 5 a_{n-2} x^n

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