IGNOU MTE 13 2018 - Solve the recurrence6an+3-13an+2+9an+1-2an = 2-n;n > 0; if a0 =

Question

Solve the recurrence
6a_n+3-13an+2+9an+1-2an = 2^-n;n > 0; if a0 = 1;a1 = 1;a₂ = 1

14 Feb 2018

Answer :

Word Count : 420

Let's carefully solve the recurrence step by step manually. The given recurrence is: $$ 6 a_n + 3 - 13 a_{n+2} + 9 a_{n+1} - 2 a_n = 2^{-n}, \quad n \ge 0 $$ with initial conditions: $$ a_0 = 1, \quad a_1 = 1, \quad a_2 = 1 $$ --- ### Step 1: Simplify the recurrence Combine like terms: $$ 6 a_n - 2 a_n - 13 a_{n+2} + 9 a_{n+1} + 3 = 2^{-n} $$ $$ 4 a_n + 9 a_{n+1} - 13 a_{n+2} + 3 = 2^{-n} $$ $$

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