Is the ‘Isha prayer valid if the worshiper joins the Tarawih prayer with the intention of completing the first two rak‘ahs and then making up the rest, but the Imam prayed four rak‘ahs with one tashahhud and performed the prostration of forgetfulness (sujud al-sahw)?

It is permissible to perform a fard (obligatory) prayer behind an imam who is performing a nafila (supererogatory) prayer, and vice versa. So, if a worshipper prays 'Isha behind an imam who is praying Tarawih, the worshipper should stand up after the imam's tasleem and complete his prayer. The prescribed way for night prayers (Qiyam al-Layl) is to pray two rak'ahs by two rak'ahs, based on the saying of the Prophet (peace be upon him): "The night prayer is two by two." The majority of scholars, however, held the view that it is permissible to pray Qiyam al-Layl four by four, interpreting the hadith as a recommendation or because it is easier for the worshipper. If the imam mistakenly stands for a third rak'ah in Tarawih, he must return, say the tashahhud, perform the tasleem, and then perform the two prostrations of forgetfulness (sujud as-sahw). If he continues and prays four rak'ahs, what the imam did is permissible according to some scholars, and his prayer is valid, and the prayer of the congregants behind him is also valid.