Organisms that later became good index fossils lived over a

wide geographic area and existed for a short geologic time.

Explanation:

Index fossils are the type of fossil that are used to define periods of geologic time.

Geologic time can simply be explained as the extensive interval of time occupied by the geologic history of Earth. And it is known to be the main factor determining the degree of rock compaction.

In this case, the index fossils are known to be widely distributed, but are limited in time span, or geologic time.