IGNOU MTE 13 2018 - Find the generating function of the recurrencean = 4an-1-4an-2+1wi

Question

Find the generating function of the recurrence
an = 4a_n-1-4a_n-2+1
with initial conditions a₀ = 1, a₁ = 1.

14 Feb 2018

Answer :

Word Count : 342

To find the generating function \( A(x) \) for the recurrence: \[ a_n = 4a_{n-1} - 4a_{n-2} + 1 \] with initial conditions: \[ a_0 = 1, \quad a_1 = 1 \] ### Step 1: Define the Generating Function The generating function is given by: \[ A(x) = \sum_{n=0}^{\infty} a_n x^n. \] Multiplying both sides of the recurrence by \( x^n \) and summing over all \( n \): \[ \sum_{n=2}^{\infty} a_n x^n = 4 \sum_{n=2}^{\infty} a_{n-1} x^n - 4 \sum_{n=2}^{\infty} a_{n-2} x^n + \sum_{n=2}^{\infty} x^n.

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