(Bamidbar 19:19) "And the clean one shall sprinkle on the unclean one": If the clean one sprinkles upon the unclean one, he becomes unclean. These are the words of R. Akiva. The sages say: Scripture is speaking only of those things which have become unclean (as being sprinkled upon, and not of things which are clean.) "on the third day and on the seventh day": Scripture comes to teach us that one who has become tamei through a dead body requires sprinkling on the third and seventh days.
You say it comes to this end, but perhaps the intent is that if he sprinkled on the third day he becomes clean on the seventh day, and if not, not? It is, therefore, written "and he shall cleanse him on the seventh day." It repeats ("on the seventh day") to indicate that it is invalid otherwise (i.e., if he does not sprinkle on the seventh day.) "and he shall wash his garments and he shall bathe in water": This tells me (that he may wash his garments) only on the seventh day.
Whence do I derive (that it is also permissible on) the eighth and ninth days? From "and he shall cleanse him on the seventh day," followed by "and he shall wash his garments and bathe in water and he will be clean in the evening."