With a correction term Forward Euler becomes Crank-Nicolson
Use the other alternative u″=−au′:
u′=−au−12aΔtu′⇒(1+12aΔt)u′=−au.
Apply Forward Euler:
(1+12aΔt)un+1−unΔt=−aun,
which after some algebra can be written as
un+1=1−12aΔt1+12aΔtun.
This is a Crank-Nicolson scheme (of second order)!