Solving a single-term recurrence relation with a variable coefficient?

Question

$a_n = 2na_{n-1}\\ a_0 = 1$

How do I solve this? Is there a characteristic equation? I found $a_1 = 2, a_2 = 8, a_3 = 48$ but I don't know what to do with that information to solve.

Please help, thanks.

Answer 1

solving the equation $q^n=2q^{n-1}$ we get $q=2$ thus the equation has the form $$a_n=C2^n$$ you will get the value of $$C$$ by inserting an initional condition.

Answer 2

Notice that $$a_n=2na_{n-1}=2n\cdot(2(n-1)a_{n-2})=2n\cdot(2(n-1)\cdot(2(n-2)a_{n-3}))=...=2^{n-1}n!a_1$$ but $$a_1=2\cdot1\cdot a_0=2$$ Hence $$a_n=2^nn!$$