Sequences & series
The idea
A sequence is an ordered list of numbers produced by a rule, and a series is the sum of its terms. Two families dominate: arithmetic sequences add a constant difference d each step, so aₙ = a₁ + (n − 1)d, while geometric sequences multiply by a constant ratio r, so aₙ = a₁ × r^(n−1). You have spotted such patterns for years; what is new is the function viewpoint — a sequence is a function on the counting numbers — plus compact formulas that jump straight to term 50 without listing 49 predecessors.
The summing formulas come from honest tricks worth understanding rather than memorizing. An arithmetic series pairs first with last: every pair has the same total, so the sum is n × (a₁ + aₙ)/2 — the number of terms times the average term. A geometric series subtracts a shifted copy of itself, leaving Sₙ = a₁(1 − rⁿ)/(1 − r). The common slip is using n where n − 1 belongs: the 20th term sits only 19 steps beyond the first, so aₙ contains (n − 1) copies of d, never n.
Worked example
A theater has 18 seats in the first row, and each row behind it has 4 more seats than the row in front. With 20 rows total, how many seats are in the back row, and how many seats does the theater hold altogether?
- Row sizes form an arithmetic sequence with a₁ = 18 and d = 4. The 20th row lies 19 steps beyond the first, so a₂₀ = 18 + 19 × 4 = 18 + 76 = 94 seats.
- For the total, pair rows from the outside in: row 1 with row 20 gives 18 + 94 = 112, and row 2 with row 19 gives 22 + 90 = 112 — every pair matches because each step in adds 4 on one side and removes 4 on the other.
- Twenty rows form 10 such pairs, so the total is 10 × 112 = 1120. The formula says the same thing: S₂₀ = 20 × (18 + 94)/2 = 20 × 56 = 1120.
- Interpret the formula: (18 + 94)/2 = 56 is the average row size, and 20 average rows of 56 seats is 1120 — the pairing trick and the average reading agree.
Answer. The back row has 94 seats and the theater holds 1120 seats in total.
Check your understanding
- Why does pairing the first and last terms of an arithmetic series always give equal pair sums, and where does the n/2 in the formula come from?
- How do you decide from a handful of terms whether a sequence is arithmetic, geometric, or neither?
- What breaks in the geometric sum formula when r = 1, and how would you compute that sum instead?
- In what sense is an arithmetic sequence a linear function and a geometric sequence an exponential one?
Build the foundations first
Sequences & series builds on these concepts. If any feel shaky, start there.