Tricky Number Sequence Puzzle

[TheRealJoeFish]

7, 7, 5, 1, 2, 4, 8...

Drumroll please:

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7

There are two ways to get it. The more mathematical way is than each number is the remainder when the sum of all of the numbers before it is divided by 9:

7+7=14==5 mod 9
7+7+5=19==1 mod 9
7+7+5+1=20==2 mod 9
...
7+7+5+1+2+4+8=34== 7 mod 9

The more intuitive way, an the way I saw how to do it first, is the old trick of repeated adding of digits until you get a single number. This is equivalent to finding the answer mod 9:

7+7=14; 1+4=5
7+7+5=19; 1+9=10; 1+0=1

And so on.

In fact, the sequence is the same if there's only one 7 at the beginning, in which case each number is (twice the previous number) mod 9. Then you get a repeating sequence of 7,5,1,2,4,8,7,5,1,2,4,8...

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Math puzzles are sort of my area of expertise; I was a "mathlete" from about 6th grade through the end of undergrad, and was on my school's Putnam team. So, what I mean to say is, I really hope I didn't make a stupid mistake in the above, because boy would I be embarrassed ;-)