Sequences and Series (Arithmetic and Geometric)

ACT Math · Axiom Academy

Arithmetic sequences add by a constant amount, geometric sequences multiply by one — master both nth-term and sum formulas before the ACT tests them. 1. Arithmetic Sequences — Add the Same Amount An arithmetic sequence adds a fixed common difference d to get from one term to the next. Take — each term is 3 more than the one before, so d = 3 . Watch the number line below: a single step of length d repeats over and over, and landing on the n th step tells you the n th term directly — no need to add d one hop at a time. nth-term formula — jump straight to any term 2. Geometric Sequences — Multiply by the Same Amount A geometric sequence multiplies by a fixed common ratio r to get the next term. Take — each term is 3 times the one before, so r = 3 . Watch the bars below grow: each bar is r times the height of the last, and that repeated scaling is exactly what the exponent in the nth-term formula is counting. nth-term formula — the exponent counts the multiplications 3. Series — Summing Without Adding Term by Term A series is just the sum of a sequence's terms, and each family has its own shortcut. For an arithmetic series, pair the first and last terms — every pair adds to the same total, so the sum is just (number of pairs) (that shared total). For a geometric series, the shrinking-or-growing pattern collapses into a clean ratio formula. Watch the pairing animation below: the first-and-last, second-and-second-to-last, and so on, all land on the identical sum.

This is the written version of the interactive lesson above. See the full ACT Math course.