Is the prayer valid if the worshiper catches up with the Imam in any Rukn with the opening Takbir, and is it obligatory to repeat it?

If one finds the Imam bowing (rakian), prostrating (sajidan), or sitting (jalisan), his opening takbeer (takbeerat al-ihram) is sufficient, and his prayer is valid. It is preferable, however, to also say the takbeer for bowing if he finds the Imam bowing. But if he finds the Imam sitting or prostrating, he should only say the takbeer for starting the prayer (takbeerat al-ihram).

If you previously said only one takbeer when joining the Imam: - If you intended it to be only the opening takbeer (takbeerat al-ihram), then your prayer is valid. - If you did not intend anything, or you intended the takbeer to be for starting the prayer in addition to bowing/prostrating/sitting, or you intended the takbeer to be only for bowing/prostrating/sitting, then the prayer is invalid in all these cases.

If your previous prayers are invalid, you must make them up immediately to the best of your ability.