Question

Consider a recurrence relation a_n = -3a_n-1 + a_n-2 for n = 1,2,3,4,… with initial conditions a₁ = 4 and a₂ = 2. Calculate a₅.

Answer 1

Consider a recurrence relation $a_n = -3a_{n-1} + a_{n-2}$   with initial conditions $a_1 = 4$ and $a_2 = 2.$ Let us calculate $a_5.$ It follows that $a_3 = -3a_{2} + a_{1}=-3\cdot2+4=-2.$ Then $a_4 = -3a_{3} + a_{2}=-3\cdot(-2)+2=8.$ We conclude that $a_5 = -3a_{4} + a_{3}=-3\cdot 8-2=-26.$

Answer: $a_5=-26.$