Intuition on Log-Concave Sequences

A sequence $(a_n)$ is said to be log-concave provided $a_i^2 \geq a_{i-1}a_{i+1}$ for all $i$.

What sorts of intuition can one have about log-concave sequences? In particular, what kind of "picture" does the property of log-concavity conjure up with regard to its graph?

What nice things happen when a sequence is log-concave? What are typical "next steps" after one has established the log-concavity of a sequence?

Any other comments related to getting a feel for log-concave sequences are most welcome.