Counts and threshold totals

A number of terms is a positive integer. An algebraic root alone is not a valid count. An AP total is quadratic in n because both the number of terms and their average change. In a positive-amount context the totals increase. Define the first event. Set up the term or sum condition, solve, retain valid integer counts, and check adjacent totals for “first exceeds”. With deposits 10,15,20,, S_n= 52n(n+3). For a total exceeding 100, S_5=100 but S_6=135, so six deposits are needed. Exam check: “exceeds” is strict. Equality does not meet it; a fractional count must not be accepted.