Notice that the question says a sequence where the number k appears k times successively.
So the sequence k=1,2,3,4,5,...... and so on.
Sequence k is an arithmetic progression.
Since the number k appears k times, the sum of the sequence up to the kth term is 1000.
Hence, 1+2+3+4+5+...+n=1000
then by using the sum of an arithmetic progression, i used it to find n
You are taking the last of the repeated numbers and calling it Tn.
T1=1
T3=2, and 1+2=3
T6=3, and 1+2+3=6
T10=4, and 1+2+3+4=10
Hence you want T1000=1+2+3+4+....+n.
However, you do not know that T1000 is the last of a repeated list of the same number.
Following CaptainBlack's advice,
T990=44
generated from the sum of terms, so it's the last 44.
Hence T991 is 45 and how many of those follow T990 ?