a) We proceed by proof by contradiction. \begin{align*} a_1+a_3 &\geq 2a_2 \\ a_2+a_4 &\geq 2a_3 \\ \vdots \\ a_7+a_9 &\geq 2a_8 \end{align*}Summing, we get $a_2+a_8 \leq 0 \implies a_2=0$ and $a_8=0$ Thus, since equality holds for the sum, equality holds for each individual statement. Thus, $a_i=0$ for all i, contradiction. (b) It is not true. Take $a_2=1$, $a_3=1.9$, $a_4=1.9^2-1$, $a_5=1.9^3-2 \cdot 1.9$, $a_6=1.9^4-3\cdot 1.9^2 +1$, $a_7=1.9^5-4\cdot 1.9^3+3\cdot 1.9$, $a_8=1.9^4-4 \cdot 1.9^2+3$