Question

Consider a recurrence relation an = an-1 - 3an-2 for n = 1,2,3,4,… with initial conditions a1 = 3 and a2 = 5. Calculate a5.

Answer 1

Let us consider a recurrence relation $a_n = a_{n-1} - 3a_{n-2}$ with initial conditions $a_1 = 3$ and $a_2 = 5.$ Let us calculate $a_5.$

It follows that $a_3 = a_{2} - 3a_{1}=5-3\cdot 3=-4.$ Then $a_4 = a_{3} - 3a_{2}=-4-3\cdot5=-19.$

We conclude that $a_5 = a_{4} - 3a_{3}=-19-3(-4)=-7.$