A interesting list

The answer is the digit sum of $n$ in base- $3$ .

Since $1000$ is $1101001$ in base $3$ , we get the answer of $1+1+0+1+0+0+1= \boxed{4}$ .

---

Note : we start counting with $0$ .

Terryfy a number once, you get a list with 3 terms. Terryfy that, you get a list with with 9 terms. Conspicuously, we are dealing with powers of nine. Our primary goal is to find the 1000th number of the list, starting with 0. Here is the solution I will use. As we see, a Terryfyied list is a sort of repetition of a list over and over again. What we can do here is to list out the lengths of every possible list that has been Terryfyied starting as a single term. Terryfy once: 3 twice: 9 three times: 27 four times: 81 five times: 243 six times: 729 We can't add a seventh time because we don't need to. We are only looking for lists below the length of 1000. The number 0 is added to bit by bit as the list increases. the 0 becomes 1 after a list of 729. 1000-729=271. After a list of 243, the 1 becomes 2. 271-243=28. After a list of 27, the 2 becomes 3. Adding a one to the end makes it become a 4 because 28-27=1 and that one is the beginning of a new list.