Gemara

Question: Why do we blow Teki'os and Teru'os while sitting (in Shacharis), and again in Musaf?

Answer (R. Avahu): This is to confuse the Satan.

(R. Yitzchak): Any year in which the Shofar was not blown on Rosh Hashanah, Meri'in (will be bad) in the end.

Question: What is the reason?

Answer: The Satan was not confused.

32b (Mishnah): We do not stop children from blowing. We engage with them until they learn.

33a (R. Elazar): This is only on Shabbos.

Support (Beraisa): We engage with them until they learn, even on Shabbos. We do not stop children from blowing on Shabbos, all the more so on Yom Tov.

Question: 'We engage with them' connotes that we tell them to blow. 'We do not stop' connotes that l'Chatchilah we do not tell them to blow!

Answer: It depends on whether or not he reached the age of Chinuch (training in Mitzvos)

Rishonim

The Rif and Rosh (1:5) bring the Gemara verbatim.

Ran (DH Lamah): Since the primary Teki'os are while standing, R. Avahu asks why we also blow sitting. The Satan is the angel of death; it is the Yetzer ha'Ra (Bava Basra 16a). 'Confusing the Satan' means humbling the Yetzer - "Im Yitoka Shofar b'Ir v'Am Lo Yecheradu". The Aruch cites the Yerushalmi that "Bola ha'Maves Lanetzach" (death will cease) when "Yitoka b'Shofar Gadol". The first time the Satan hears the Shofar, he is not so panicky. When he hears us blow again, he is sure that this is the great Shofar, and that his end is near. He trembles, is confused and has no opportunity to prosecute. It seems that even though it would be fitting to blow first the primary Teki'os with the Berachos, if so people might not be concerned for the other Teki'os, which are only to confuse the Satan.

Ran (DH v'Omar (1)): BaHaG says that R. Yitzchak's teaching (any year in which the Shofar was not blown on Rosh Hashanah will be bad in the end) does not apply when Rosh Hashanah was on Shabbos and we did not blow due to the decree.

Rif: On Shabbos we engage with a child who reached the age of Chinuch. For a child below Chinuch it is permitted on Yom Tov, but not on Shabbos. Some say oppositely.

Hagahos Gra (34a): The Rif's text omits 'on Shabbos, all the more so on Yom Tov'.

Ran (DH Omar): The Yerushalmi says that we permit a child who reached Chinuch. It is permitted on Shabbos only if it is Rosh Hashanah.

Question: Why must we stop younger children? We need not prevent a child from transgressing!

Answer (Ran DH v'Hani): We need not stop children if the matter will not become known and lead to transgression. Here, people will hear the Teki'os, and (adults) might come to blow on Shabbos (outside of Beis Din).

Ran (DH v'Chosav): R. Efrayim and Ba'al ha'Ma'or say that the Heter for children on Shabbos was only when Beis Din blew on Shabbos. Nowadays even children may not blow. The Rif (brings only what applies nowadays; he) holds that even nowadays we may blow in a fixed Beis Din. The Ramban says that since all the Ge'onim bring the Heter, it applies even when we do not blow in Beis Din.

Rambam (Hilchos Shofar 3:10): The custom regarding Teki'os of Rosh Hashanah b'Tzibur is that after Kri'as ha'Torah everyone sits, and one stands and blesses 'Asher Kidshanu b'Mitzvosav v'Tzivanu Lishmo'a Kol Shofar' and Shehecheyanu and blows 30 Teki'os due to the Safek (about what is "Teru'ah"). Later, in Musaf, when the Shali'ach Tzibur finishes the fourth Berachah he blows Teki'ah, three Shevarim, Teru'ah, Teki'ah. After the fifth Berachah he blows Teki'ah, three Shevarim, Teki'ah. After the sixth Berachah he blows Teki'ah, Teru'ah, Teki'ah.

Magid Mishnah: (This is like the Ge'onim.) R. Yakov enacted to blow Shevarim-Teru'ah for each of the middle Berachos. The Ramban and Rashba say that one should do like the Ge'onim and the Rambam.

Rambam (12): Letter of the law, for each Berachah we should blow each Bava (Teru'ah with a Teki'ah before and after) three times, like (the first time,) when we were sitting. However, since we were already Yotzei the Safek while sitting, we do not exert the Tzibur to repeat all of them along with the Berachos. It suffices to blow one Bava for each Berachah, so the Tzibur will hear them along with the Berachos. There is no custom regarding individuals. Whether or not one heard along with the Berachos, whether he was standing or sitting he was Yotzei.

Rambam (2:7): On Shabbos that is not Rosh Hashanah we do not stop children under the age of Chinuch from blowing, in order that they will learn. An adult may engage with children in order that they will learn on Yom Tov, whether or not they reached Chinuch. This is because Teki'ah is only Shevus.

Rebuttal (Ra'avad): R. Elazar taught that on Shabbos we may engage with them. The Rambam never permits this! Rather, on Shabbos right before Rosh Hashanah one may engage with children that reached Chinuch.

Answer (Magid Mishnah): Indeed, the Rambam permits engaging 'On Yom Tov'

Rosh (4:10): We are Yotzei the Mitzvah with the first 30 Teki'os after Kri'as ha'Torah. We blow later merely to fulfill hearing them with the Berachos of Musaf. Our custom is to blow Shevarim-Teru'ah (with a Teki'ah before and after) for Malchuyos, Shevarim for Zichronos, and Teru'ah for Shofaros. R. Tam questioned the inconsistency. Rather, he enacted to blow Shevarim-Teru'ah for each. One is Yotzei, for the extra sounds do not interrupt. Perhaps our custom holds that mid'Oraisa, one needs only one "Teru'ah" (and we are unsure if this is what we call Shevarim or Teru'ah or Shevarim-Teru'ah) with a Teki'ah before and after it. Mid'Rabanan, we require three such Teru'os. After being Yotzei all the Sefekos and the mid'Rabanan addition, we blow again with the Berachos enough to ensure that we blow what the Torah required.

Rosh (11): Once, the Toke'a erred. The Tzibur told him to repeat sounds he had blown. Rav Elyakim bar Yosef and R. Eliezer bar Noson said that this was unnecessary; the Toke'a transgressed Shevus by blowing extra Teki'os.

Korban Nesan'el (7): The Shevus is a decree lest one fix a musical instrument.

Question (Tosfos DH v'Tok'im): The Toke'a transgresses Bal Tosif for blowing extra!

Suggestion: Since he was already Yotzei, it is not considered the time of the Mitzvah, therefore Bal Tosif does not apply.

Rejection: A Kohen who adds a Berachah after Birkas Kohanim transgresses Bal Tosif. Even though he was Yotzei, it is still considered the time of the Mitzvah, since if he would chance upon another Tzibur he could bless them. Also here, the Toke'a could blow for another Tzibur!

Answer (Tosfos): Bal Tosif does not apply to repeating a Mitzvah. A Kohen would not transgress if he repeated the Berachos of Birkas Kohanim. One would not transgress for extra Zerikos on the same corner of the Mizbe'ach.

Poskim

Shulchan Aruch (596:1): At the end of Tefilah we blow a long Teru'ah without a Teki'ah.

Source (Kitzur Piskei Rosh 4:10): Rav Amram Gaon says to do so to confuse the Satan.

Mishnah Berurah (1): This is so the Satan will not prosecute that Yisrael are going to eat and drink because they do not fear the judgment. Our custom is not to blow an extra Teru'ah. Rather, we make the last of the 100 blows extra long.

Rema: Some places blow another 30 sounds.

Taz (1): It seems that these 30 are to fix the sin of people who might not have listened properly before.

Beis Yosef (OC 585 DH u'Mah she'Chosav Rabeinu Perush): The SMaG says that even if the Satan would not fear that this is Shofar ha'Gadol, in any case it reminds him of his end and inhibits his prosecution.

Bach (DH Meri'in): The Beis Yosef explains 'Meri'in in the end' to mean that we will need to blow Teru'ah over calamities. The Maharshal says that it means 'it will be bad'.

Prishah (5): It is better to take the right horn of a ram, for it is bigger (and more easily confused with Shofar ha'Gadol).

Mishnah Berurah (581:24): We do not blow the Shofar on Erev Rosh Hashanah to distinguish between optional Teki'os of Elul to obligatory Teki'os of Rosh Hashanah. Minhagim says that it is to confuse the Satan (so that he will think that the Teki'os of Rosh Hashanah are from the Shofar ha'Gadol).

Rema: After this adults do not blow again needlessly.

Magen Avraham (2): Ra'avan forbids to move a Shofar after blowing all he needs to hear. Even one who must Toke'a on Yom Tov Sheni should not practice on Yom Tov Rishon, for nowadays we know that mid'Oraisa the second day is Chol.

Rebuttal (Taz 2): The Gemara says that Teki'ah is a Chachmah, not a Melachah. It is forbidden on Shabbos lest one carry in Reshus ha'Rabim, but there is no decree when Rosh Hashanah is not on Shabbos. If a Shofar was Muktzeh after one was Yotzei, the same should apply to Lulav!