Consider a recurrence relation an = -3an-1 + n for n = 1,2,3,4,… with initial conditions a1 = 3. Calculate a3.

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Recurrence relation:

$a_{n} = -3a_{n-1} + n$

Initial condition:

$a_{1} = 3$

Use recurrence relation for $n = 2$ :

$a_{2} = -3a_{1} + 2 = -3 * 3 + 2 =-9 + 2 = -7$

Use recurrence relation for $n = 3$ :

$a_{3} = -3a_{2} + 3 = -3 * (-7) + 3 = 21 + 3 = 24$

Answer: $a_{3} = 24$