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How is an index fossil useful to geologists in establishing the age of the rock layer in which they find it?

Answer
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Hint: The fossils which are used to define and identify geological periods are known as index fossils. Short vertical range, wide geographic distribution and rapid evolutionary trends are the characteristics of index fossils.

Complete answer: The relative age of a rock can be determined if there is an index fossil present in the rock. A fossil must be widely distributed and it must also represent a type of organism that existed for a brief time period, if all these conditions are present then the fossil can be useful as an index fossil.
Index fossils can be a plant or an animal and are preserved in the rock record of the earth. It is characteristic of a particular span of geologic time or environment. Other characteristics of an index fossil are easily recognizable or distinct, abundant, a wide geographic distribution and a short range through time.
Ammonites which were common during the Mesozoic Era are an example of index fossils. It is believed that index fossils vanished from the earth in a known extinction event or lived for a short period of time.
Human Beings can also become good index fossils in the future. Humans bury their dead and we have hard parts. Therefore at some time in the distant future we may become index fossils.

Note: Note: Brachiopods, Graptolites, Trilobites are all examples of index fossils. Mammals were widely used to date deposits in the terrestrial sediments of the Cenozoic Era which began about 65.5 million years ago.