I'm interested in showing by induction that if I have a product of geometric series and a product of binomials that if for some $x^i$ which has coefficient $a_i$ and I know $a_i$ is positive then $a_{i-1}$ is also positive. Here is my equation: $$(1+x+x+...x^{k_0})(1+\sum_{r=1}^{{k_1}}(x+c_1)^r) (1+\sum_{r=1}^{{k_2}}(x+c_2)^r)\cdot \cdot \cdot(1+\sum_{r=1}^{{k_m}}(x+c_m))-{2\prod_{j=1}^{k_j}(x+c_j)^{k_j}}=0$$
So in order to show this I'm unsure what my base case would be and also is this going to be backwards induction?