linear algebra - Find the characteristic polynomial through the induction

$A = \begin{bmatrix} 0 & 0 & \dots & 0 & a_{0} \\ 1 & 0 & \dots & 0 & a_{1} \\ \ 0 & \ddots & \ddots & \vdots & \vdots \\ 0 & \dots & 1 & 0 & a_{n-1} \\ 0 & \dots & 0 & 1 & a_{n} \\ \end{bmatrix}$

I need to find the characteristic polynomial through the induction. I guess we should use the formula $\det \left(A-\lambda I\right)$ and then expand the first row, since we have a lower diagonal matrix, the determinant of this matrix will be equal to 1.

I'm having problems with setting up the induction formally using the definition of characteristic polynomial and finding the characteristic polynomial.