Arithmetic & Geometric Progressions

• A progression (sequence) is an ordered list of numbers following a rule; a series is what you get when you add those terms up.
• Sigma notation (Σ) is shorthand for "add up all these terms" — useful for understanding series, though it won't appear directly in your exam.
• In an arithmetic progression, you add the same number (the common difference, d) each time. Formula: u_n = a + (n − 1)d
• The sum of an arithmetic series uses S_n = ½n{2a + (n − 1)d} or S_n = ½n{a + l}
• In a geometric progression, you multiply by the same number (the common ratio, r) each time. Formula: u_n = ar^{n − 1}
• The sum of a geometric series uses S_n = a(1 − rⁿ)/(1 − r), and if |r| < 1, the series converges to a sum to infinity: S_∞ = a/(1 − r)
• Questions frequently combine these formulas with simultaneous equations (two unknown terms) or a quadratic that needs solving.

What is a progression?

A progression (also called a sequence) is simply an ordered list of numbers that follows a rule. Think of it like a line of dominoes falling — each one knows exactly what comes next, because there's a consistent rule connecting them.

For example: 1, 3, 5, 7, 9, … is a sequence with the rule "start at 1 and add 2 to each number." Every number in this list is called a term.

Rather than writing "the 3rd number in the sequence," mathematicians use subscript notation with the letter u. So for the sequence above:

Each term can be found by substituting the term number into a formula for the n-th term — this is the whole game of this chapter: find that formula, and you can find any term instantly, without listing everything before it.

What is a series?

A series is what happens when you stop listing the terms and start adding them up. For the sequence 1, 3, 5, 7, … the associated series is 1 + 3 + 5 + 7 + …

We write S_n to mean "the sum of the first n terms":

Sigma notation (Σ) — reading it, not testing it

Σ what which

r k