Praise be to Allah.
Praying Taraweeh at home is permissible and there is nothing wrong with it, but offering the prayer in congregation in the mosque is better.
The scholars of the Standing Committee were asked:
The month of Ramadaan has come and it is time to pray Taraweeh. Should I go to the mosque or pray in my house? I am not an imam, rather I am one of the congregation, but I like to recite Qur’aan, and I prefer to recite rather than to listen. If I pray at home, is there any sin on me? I am referring to Taraweeh prayer only.
There is no sin on you if you pray in your house because it is naafil. But praying with the imam in the mosque is better, following the example of the Prophet (peace and blessings of Allaah be upon him) and his companions (may Allaah be pleased with them), because when the Prophet (peace and blessings of Allaah be upon him) led his companions in prayer at night until the end of the first third of the night, one of them said to him: We wish that you would lead us in naafil prayer for the rest of the night. He said: “Whoever prays qiyaam with the imam until he finishes, it is as if he spent the whole night in prayer.” Narrated by Ahmad (5/159) and the authors of al-Sunan with a hasan isnaad from the hadeeth of Abu Dharr (may Allaah be pleased with him). End quote.
Fataawa al-Lajnah al-Daa’imah (7/201-202).
The basic principle concerning Taraweeh prayer is that it is one of the prayers in which Qur’aan is to be recited out loud, because this is proven from the Sahaabah (may Allaah be pleased with him) at the time of ‘Umar ibn al-Khattaab (may Allaah be pleased with him), when Ubayy ibn Ka’b and others would lead the people in prayer and recite at length.
But reciting out loud in prayers in which Qur’aan is recited out loud, and reciting silently in prayers in which Qur’aan is recited silently, is something that is recommended in prayer for the imam, but it is not obligatory, as is the view of the majority of scholars among the Maalikis, Shaafa’is and Hanbalis.
It says in al-Mawsoo’ah al-Fiqhiyyah (16/188):
The majority of fuqaha’ are of the view that reciting out loud in prayers in which Qur’aan is recited out loud, and reciting silently in prayers in which Qur’aan is recited silently, is one of the Sunnahs of prayer. The Hanafis are of the view that it is obligatory to recite out loud in prayers in which Qur’aan is recited out loud, and to recite silently in prayers in which Qur’aan is recited silently. End quote.
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said in Noor ‘ala al-Darb (al-Salaah/218):
Reciting out loud in prayers in which Qur’aan is recited out loud is not obligatory, rather it is preferable. If a person recites silently in a prayer in which it is prescribed to recite out loud, the prayer is not invalid, because the Prophet (peace and blessings of Allaah be upon him) said: “There is no prayer for the one who does not recite the Essence of the Qur’aan (i.e., al-Faatihah).” And there is no stipulation that this recitation should be done out loud or silently. So if a person recites what he is required to recite, whether it is silent or out loud, he has done what is required. But it is better to follow the Sunnah in prayers in which Qur’aan is recited out loud, such as the first two rak’ahs of Maghrib and ‘Isha, and Fajr prayer, and Jumu’ah and the Eid prayers, and prayers for rain (istisqa’) and Taraaweeh prayer and so on, where it is customary. If a person deliberately does not recite out loud when he is leading the prayer, his prayer is still valid, but it is lacking. As for the person who is praying alone, if he offers a prayer in which Qur’aan is recited out loud then he has the choice between reciting out loud or silently, so he should decide which will make him focus better, and do that. End quote.
It says in Fataawa al-Lajnah al-Daa’imah li’l-Ifta’ (6/392):
It is proven that the Prophet (peace and blessings of Allaah be upon him) used to recite Qur’aan out loud in the two rak’ahs of Fajr, and in the first two rak’ahs of Maghrib and ‘Isha’ prayer, so reciting out loud in these prayers is Sunnah, and it is prescribed for his ummah to follow his example because Allaah says (interpretation of the meaning): “Indeed in the Messenger of Allaah (Muhammad صلى الله عليه وسلم) you have a good example to follow for him who hopes for (the Meeting with) Allaah and the Last Day, and remembers Allaah much” [al-Ahzaab 33:21]. And it is proven that the Prophet (peace and blessings of Allaah be upon him) said: “Pray as you have seen me praying.”
So if he recites silently in prayers where Qur’aan should be recited out loud, he is forsaking the Sunnah, but his prayer is not invalidated by that. End quote.
To sum up: your prayer is valid and you do not have to do anything else.
And Allaah knows best.