Should a latecomer count an extra rak’ah done with the imam?
Question

                If a latecomer joins the prayer with the imam in the second rak’ah, and the imam forgets and does an extra rak’ah, what should the latecomer do? Should he say the salaam with the imam or stand up after the imam says the salaam and do another rak’ah?.
Praise be to Allah.
    He should say the salaam with the imam, because his prayer is 
    complete. With regard to the imam, he is excused for this extra rak’ah. 

    Shaykh Ibn ‘Uthaymeen was asked: If the imam prays five 
    rak’ahs by mistake, what is the ruling on his prayer and the prayer of those 
    behind him? Should a latecomer count this extra rak’ah? 

    He replied: if the imam prays five rak’ahs by mistake, his 
    prayer is valid, and the prayer of those who followed him by mistake or out 
    of ignorance is also valid. 

    But if a person knew that this was something extra, when the 
    imam stood up for the extra rak’ah, he should have sat and said the salaam, 
    because in this case he thought that the prayer of his imam was invalid, 
    unless he feared that his imam stood up for the extra rak’ah because he did 
    not recite al-Faatihah properly (for example) in one of the two rak’ahs. In 
    that case he should wait and not say the salaam. 

    With regard to the latecomer who joined the imam in the 
    second or subsequent rak’ah, this extra rak’ah counts for him. So if he 
    joined the imam in the second rak’ah, for example, he should say the salaam 
    with the imam who prayed an extra rak’ah. If he joined in the third rak’ah, 
    then he should do another rak’ah after the imam says the salaam following 
    the additional rak’ah. That is because if we say that the extra rak'ah does 
    not count for the latecomer, he would have to add another rak’ah 
    deliberately, which would mean that the prayer was invalid. As for the imam, 
    he is excused for the extra rak’ah, because he forgot (and did it by 
    mistake), so his prayer is not invalidated. 

    Majmoo’ Fataawa Ibn ‘Uthaymeen, 
    14/20.