I am familiar with the origin and reasoning of the "regular" Tekiah blast that surrounds the shevarim, teru'ah and shevarim-teru'ah sounds.
What is the origin of the teki'ah gedolah in terms of its use on Rosh Hashanna? I recall that that there are various verses that allude to a "long" Shofar sound such as the one heard on Mt. Sinai, the one used for yovel (Jubilee) and the sound of the "great shofar" announcing Mashiach.
But what is the origin of using it as part of the Rosh Hashanna shofar service? I don't recall this mentioned in the Talmud anywhere.
It seems to be done as the last teki'ah blast of some type of "grouping" such as the last blast of the teki'ot d'meyushav - the group of blasts done before Musaph Amidah. Is this its purpose? To announce the ending of a grouping? Or is there some other purpose to it?