Sequences and series

A sequence is an ordered list of numbers: 2, 5, 8, 11, … A series is what you get when you add a sequence up. Two patterns cover most of what you'll meet — and both have formulas that beat adding term by term.

Two ways a sequence can grow

• Arithmetic — each term adds a fixed common difference d. 2, 5, 8, 11 has d = 3. The nth term is aₙ = a₁ + (n − 1)d. Plotted, the terms land on a straight line.
• Geometric — each term multiplies by a fixed common ratio r. 1, 2, 4, 8 has r = 2. The nth term is aₙ = a₁ · r⁽ⁿ⁻¹⁾. Plotted, the terms curve away exponentially.

The test: subtract consecutive terms — constant? arithmetic. Divide consecutive terms — constant? geometric.

Summing without slogging

Adding the first n terms has a closed form for each type:

The arithmetic formula is the famous Gauss trick: pair the first term with the last, the second with the second-to-last — each pair sums to (a₁ + aₙ), and there are n/2 pairs.

The infinite geometric series

Here's the surprise that points straight at calculus. If |r| < 1, each term is a shrinking fraction of the one before — and adding infinitely many of them still gives a finite total:

S = a₁ / (1 − r), for |r| < 1

So 1 + ½ + ¼ + ⅛ + … = 1 / (1 − ½) = 2. Infinitely many positive numbers, adding to exactly 2.

Why this matters

That an endless sum can land on a finite value is the seed of calculus: integrals are infinite sums of infinitely thin pieces. Sequences and series also drive compound interest, loan amortization, and the way computers approximate functions like sin and eˣ.