Linear multistep methods have even order of convergence

Question

A linear multistep method $$ \sum_{j=0}^n\alpha_jx_{n-j}=h\sum_{i=0}^n\beta_if(x_{n-j},t_{n-j}) $$ is called symmetric, if $$ \alpha_{n-j}=-\alpha_{j}, \quad \beta_{n-j}=\beta_{j}, \quad j=0,...,n $$ Now how can one show that the order of convergence for symmetric linear multistep methods is always an even number?

Answer 1

Formulate the order condition for the symmetric test functions $x(t)=(t-nh/2)^m$ to see that the conditions for $m$ even are trivially satisfied due to symmetry.