Common Index Counts

Let $\{a_n\}$ and $\{b_n\}$ be two distinct infinite sequences, neither constant. Define $M = \{k \mid a_k = b_k,\ k \in \mathbb{N}^*\}$. Determine which of the following statements are correct, with justification:

1. if $\{a_n\}$ and $\{b_n\}$ are both arithmetic, then $M$ has at most $1$ element; 2. if $\{a_n\}$ and $\{b_n\}$ are both geometric, then $M$ has at most $2$ elements; 3. if $\{a_n\}$ is arithmetic and $\{b_n\}$ is geometric, then $M$ has at most $3$ elements; 4. if $\{a_n\}$ is increasing and $\{b_n\}$ is decreasing, then $M$ has at most $1$ element.