Suppose I wanted to solve the following for $a[1], a[2], a[3]$ in the Mathematica code:
(a[1]+a[3])*t+(a[1]+2*a[2])*t^2=0
with $a[1] = 1$ where $t$ is just a variable and the equation above is identically 0. How would I automate this in Mathematica to get $a[3] = -1, a[2] = -1/2$?