Let $Y_1, Y_2, . . .$ be an adapted sequence, and let $c_n\in \mathcal{R}$, $n \ge 1$.
(a) Suppose that $E(Y_{n+1} | F_n) = Y_n + c_n$. Compensate suitably to exhibit a martingale.
(b) Suppose that $E(Y_{n+1} | F_n) = Y_nc_n$. Compensate suitably to exhibit a martingale.
I thought that I can choose for:
(a) $a_{n+1}-a_n=c_n$;
(b) $b_{n+1}=c_nb_n$
If it is true how can I show it is a martingale?
The idea in your answer is good, but you have to be more explicit.
For (a), you want a sequence $(a_n)$ such that defining $Z_n:=Y_n+a_n$, the sequence $(Y_n,\mathcal F_n)$ is a martingale. And indeed, after having computed $\mathbb E\left[Z_{n+1}\mid\mathcal F_n\right]-Z_n$, we see that we need $a_{n+1}-a_n=c_n$.
Same thing for (b), except that this time, $Z_n:=b_nY_n$.
In both cases, the $a_n$ and $b_n$ can be written explicitely in terms of $c_n$'s.