Gemara

Beraisa: If a city has 1500 Raglei (people fitting to go to war) and nine people died in three days, one after the other, this is Dever (for which we fast and Masri'a). If they died in one or four days, it is not Dever. If a city has 500 Raglei and three people died in three consecutive days, this is Dever. If they died in one or four days, it is not Dever.

A case occurred, the specified number died in one day, and Rav Nachman bar Rav Chisda decreed a Ta'anis.

Rav Nachman bar Yitzchak: This is like R. Meir, who says that if intermittent gorings make a Mu'ad, all the more so closely spaced gorings do!

Bava Kama 24a (Mishnah - R. Yehudah): An animal is Mu'ad after witnesses testify that it gored on three different days;

R. Meir says, it is Mu'ad after witnesses testify three times that it gored (even on one day).

Question: What is R. Meir's reason?

Answer: If an ox becomes Mu'ad after intermittent gorings, all the more so when the gorings are close together!

Rishonim

Rosh (Ta'anis 1:1): In summer if one mentioned rain he goes back. The Yerushalmi says that if one does not know what he mentioned, within 30 days (of changing) we assume that he said like he was used to.

Hagahos Maimoniyos (Rambam (Hilchos Tefilah 10:5): If one wants to avoid praying again when in Safek (about what he mentioned) he should say (right after we change) from 'Atah Gibor' until Mechalkel Chayim as many times as we say it in 30 days (and then the Chazakah is that he said the proper text). The same applies to Birkas ha'Shanim. The Gemara (Bava Kama 24a) says regarding Shor ha'Mu'ad that if a Chazakah is established over a long period of time, all the more so it is established (by the same number of occurrences) in a shorter period. The same applies here.

Poskim

Shulchan Aruch (OC 114:8): In summer if one is in doubt whether or not he mentioned Morid ha'Geshem, if it is within 30 days (of when we ceased to mention rain) the Chazakah is that he mentioned it and he must go back.

Beis Yosef (DH v'Chosav): Sefer Ohel Mo'ed is unsure whether or not one is in doubt whether or not he mentioned or requested rain within 30 days (after we cease or begin) should go back even if he could hear the Shali'ach Tzibur's repetition. Many Ge'onim say that regarding any mistake, if one will be there (to hear the repetition) from the beginning to the end he is Yotzei.

Rema: Likewise for us who do not mention dew in summer, if one is in doubt whether or not he mentioned Morid ha'Geshem in winter, within 30 days of when we started to mention the Chazakah is that he did not mention and he must go back; after 30 days he does not return.

Mishnah Berurah (37): Many hold that '30 days' alludes to the number of constant Tefilos (i.e. excluding Musaf) that a person says in 30 days, i.e. 90. We say Morid ha'Geshem (three times daily, and four times on days with Musaf) in 90 Tefilos in less than 30 days. We do not say Ten Tal u'Matar on Shabbos and Yom Tov, so we need more than 30 days for 90 repetitions. Others say that the Chazakah always changes after 30 days, regardless of how many Tefilos were said. We are lenient about Safek Berachos, therefore one does not repeat Shemoneh Esre if 30 days or 90 Tefilos have passed.

Note: Elsewhere, when the Mishnah Berurah is indecisive about a Machlokes ha'Poskim he says that one should pray again on condition that if he is not obligated it is a Nedavah (see Bi'ur Halachah 117:5 DH ha'Tzerichim). It is not clear why he does not say so here, or that one should try to be Yotzei by listening to the repetition of the Shali'ach Tzibur.

Mishnah Berurah (38): If at the beginning of Tefilah one intended to make the proper insertion or omission, and long after prayer he is unsure whether or not he did so, he need not go back. If the doubt arose immediately after prayer he goes back.

Kaf ha'Chayim (48): He goes back only if the doubt arose during the Tefilah itself.

Kaf ha'Chayim (52): Some say that if a person knows that his Chazakah did not change after 30 days, when in doubt he should pray again Nedavah or try to be Yotzei by listening to the repetition. Some say that if a person is sure that his Chazakah changed, even before 30 days, he need not go back when in doubt.

Kaf ha'Chayim (54): One should put a reminder in his Sidur for insertions or omissions that must be said. This is even for Morid ha'Geshem for we who normally mention dew, even though we are Yotzei even if we say like we were accustomed to.

Shulchan Aruch (9): If on the first day of Pesach one begins the Berachah 'Atah Gibor...' until Morid ha'Tal 90 times, corresponding to saying it three times a day for 30 days, from then on if he is in doubt whether or not he mentioned rain the Chazakah is that he did not mention it and he does not go back.

Tur: The RaM me'Rotenburg used to do this. R. Peretz says that elders of France did not do so because this is different than Shor ha'Mu'ad. If an ox gores three times within a shorter period this shows that all the more so it is Muchzak to gore. However, prayer depends on what a person is used to saying. We do not say that repetitions in a shorter period establish the Chazakah.

Beis Yosef (DH veha'RaM): The Avudraham says that R. Peretz argues because in Tefilah we say Birkas Avos first, so it does not suffice to say from Atah Gibor until Meshiv ha'Ru'ach to change his habit of prayer. Mahari Avuhav says that if an ox gores within a shorter period this is a better proof of its bad nature; we do not attribute it to chance. Speaking habits are not changed as well in a shorter period of time. The Rosh and SMaK hold like the RaM me'Rotenburg, so we follow them.

Question (Magen Avraham 13): The Halachah follows R. Yehudah, who disagrees with R. Meir's Kal va'Chomer. The Shulchan Aruch (OC 576:2) says that we fast due to Dever if three people died in three days, but not if all died in one day. This is like R. Yehudah!

Answer (Magen Avraham ibid.): A spasm could cause an ox to gore repeatedly in one day, so perhaps it is not Mu'ad; a problem in the air could cause many deaths in one day, so perhaps it is not really Dever. But if one changes his speaking habits, the change becomes ingrained better if it is over a short period of time. Regarding this we hold like R. Meir.

Question (Nesiv Chayim): The Magen Avraham means that R. Yehudah agrees to the Kal va'Chomer when there is no reason opposing it. If so, why does the Gemara (Bava Basra 28b) ask 'according to R. Meir, who says the Kal va'Chomer...'?

Note: The Magen Avraham says 'regarding this we hold like R. Meir'. This connotes that R. Yehudah argues also here, just that the Halachah follows R. Meir here. However, it is not clear what is the source for this.

Rebuttal (Taz 13): The Halachah follows R. Yehudah, who learns from Zavah (three sightings over three days make her a Zavah, but not three sightings in one day). He learns from there to Shor ha'Mu'ad and other places! Therefore, one should not rely on the RaM me'Rotenburg.

Darchei Moshe (2): The Yerushalmi mentioned 30 days not only regarding mention, but also regarding request of rain. In 30 days there are more than 90 mentions, but less than 90 requests! The RaM me'Rotenburg infers that the Yerushalmi teaches that it depends on the number of repetitions. It says 30 to allude to 90 repetitions, for a normal day has three.

Rebuttal (Magen Avraham 12): The Yerushalmi does not discuss request. Hagahos Maimoniyos (10:5) does not mention 90, rather, the number of Tefilos in 30 days.

Gra (DH Im): The fact that the Yerushalmi mentions 30 days regarding mention and request, even though there are more mentions than requests in this time, shows that it depends on 30 days, not on the number of repetitions. This shows that R. Peretz is correct.

Question (Teshuvas Rema Sof Siman 83 DH v'Zos): On Shabbos we also mention rain in Musaf, so there are more than 90 mentions in 30 days!

Answer (Taz 13): Only the three Tefilos that a person is accustomed to every day help to change the Chazakah. Likewise, 30 days changes the Chazakah regarding Ten Tal u'Matar, even though this does not apply on Shabbos. It seems that we are not concerned if the person missed praying a day or two; otherwise, the RaM (me'Rotenburg) would have discussed this.

Rebuttal (Mishnah Berurah 37): There are Acharonim who say that we are concerned for missed days.

Mishnah Berurah (42): If one said it 45 times (at once), after 15 days the Chazakah will be that he said it correctly.

Mishnah Berurah (41): The Chasam Sofer says that l'Chatchilah one should say it 101 times.

Rema: (We do not mention dew in summer, so in winter if one said like he is used to he must go back.) Likewise for us, if one says on Pesach (from Atah Gibor) until Mechalkel Chayim without Meshiv ha'Ru'ach u'Morid ha'Geshem, or on Shemini Atzeres from Atah Gibor until Meshiv ha'Ru'ach u'Morid ha'Geshem 90 times, from then on the Chazakah is that he says correctly.

Rebuttal (SHLaH Maseches Tamid 69): Repeating a text with an insertion is an action; it habituates a person to say that insertion. But passivity, i.e. repeating a text and omitting (something that he used to say) does nothing (to change his Chazakah).

Note: Seemingly, this reasoning applies even if the repetitions were done over the course of 30 days, or even six months. If so, at the beginning of winter why is there a Chazakah that one omitted rain?

Mishnah Berurah: It is proper to be concerned for the SHLaH's opinion.

Magen Avraham (14): (When saying without mentioning rain or dew,) one must say until and including Mechalkel Chayim.

Mishnah Berurah (39): Shulchan Shlomo says that one should start from 'Mechayeh Mesim Atah' to avoid needless mention of Hash-m's name. One should not start from Atah Gibor and makes a substitution for His name, lest he will become used to this and do the same in Tefilah!