Question

Solve the single-term recurrence relation with variable coefficient: \(a_n = 2na_{n-1}; a_0 = 1\). I found \(a_1 = 2, a_2 = 8, a_3 = 48\), but what to do with this info? Is there a "characteristic equation" for this form?

Answer 1

@dan815

Answer 2

What, he would know?

Answer 3

try
\[a_n = 2^n n!a_0\]

Answer 4

That does seem to work, thanks... but how did you get that?

Answer 5

\[\begin{align}
a_n &=2na_{n-1}\\~\\
&=2n(2(n-1)a_{n-2}) = 2^2n(n-1)a_{n-2}\\~\\
&= 2^2n(n-1)(2(n-2)a_{n-3}) = 2^3 n(n-1)(n-2)a_{n-3}\\~\\
&=\cdots\\~\\
& = 2^nn! a_0
\end{align}\]