three generations; three periods
Refers to three successive generations of a family, or three historical periods/eras. Often used in contexts like family businesses, dynasties, or historical timelines.
This shop is a long-established store that has continued for three generations.
This shop is a long-established business that has lasted three generations.
The results of research spanning three generations were announced.
third generation
Specifically the third generation in a sequence, such as a third-generation owner, immigrant, or product version.
He is the third-generation company president.
He is the third-generation owner of the long-established inn.
They have run a farm across three generations.
代々 is an adverb meaning 'for generations, from generation to generation' with no fixed number, while 三代 specifically means three generations.
世代 is a general term for 'a generation' as an age group or cohort, whereas 三代 counts three specific successive generations in a lineage.
Compound of 三 (three) and 代 (generation, era). The reading さんだい is the standard on'yomi combination.