Maximum number of roots

Question

For non-zero $a_1,a_2,\dots,a_n$ and for $α_1,α_2,\dots,α_n$ such that $α_i\ne α_j$ for $i\ne j$, show that the equation $$a_1e^{α_1x}+a_2e^{α_2x}+\dots+a_ne^{α_nx}=0$$ has at most $n-1$ real roots.

The original image of the problem

I just don't have a clue how to start.

I tried with Rolle's, and I tried differentiation...but no avail

Answer 1

note that $a_1e^{\alpha_1 x}+\ldots +a_ne^{\alpha_n x}= 0 \iff 1 + \frac{a_2}{a_1}e^{(\alpha_2-\alpha_1) x}+\ldots +\frac{a_n}{a_1}e^{(\alpha_n-\alpha_1) x}=0$. Rolle along with induction should do the trick