Formula sheet: Sequences and Series
Sequences and Series16 min readLesson 1 of 6
1.1 Vocabulary
A sequence is an ordered list of numbers following a rule. The position of a term matters, so the first term, second term and so on are different objects even when two values happen to be equal. A sequence with finitely many terms is finite; otherwise it is infinite. The associated series is the indicated sum of the terms, and it is finite or infinite together with the sequence.
A progression is a sequence whose terms follow a fixed pattern. The two patterns in school mathematics are the arithmetic progression, where a constant difference separates consecutive terms, and the geometric progression, where a constant ratio separates them. The chapter works mainly with geometric progressions, but the arithmetic mean and some word problems need the arithmetic progression formulas, so they are listed first.
1.2 Arithmetic progression recap
In an arithmetic progression with first term a and common difference d, the nth term is a plus (n minus 1) times d. The sum of the first n terms is n by 2 times (2a plus (n minus 1)d), or equivalently n by 2 times (first term plus last term).
The arithmetic mean of two numbers is their average, and three numbers are in arithmetic progression exactly when the middle one is the arithmetic mean of the other two. This single fact drives the AM-GM relation at the end of the chapter.
1.3 Geometric progression
In a geometric progression each term after the first is obtained by multiplying the previous term by a fixed non-zero number, the common ratio r. The nth term is a times r to the (n minus 1). The sum of the first n terms is a times (r to the n minus 1) divided by (r minus 1) when r is not 1, and simply n times a when r equals 1.
| Quantity | Formula |
|---|---|
| nth term | a times r to the (n minus 1) |
| sum, r not 1 | a(r to the n minus 1) by (r minus 1) |
| sum, r equal to 1 | n times a |
| sum when the ratio is below 1 | a(1 minus r to the n) by (1 minus r) |
The two forms of the sum are the same expression with numerator and denominator multiplied by minus 1. Use the form that keeps the numbers positive, because it reduces sign errors.
1.4 Geometric mean and the AM-GM relation
For two positive numbers, the geometric mean comes from multiplying them and then taking the square root. Three positive numbers are in geometric progression exactly when the middle number is the geometric mean of the other two. For more than two numbers, the geometric mean is the nth root of the product, but the chapter only needs the two-number case.
The arithmetic mean is never smaller than the geometric mean for positive numbers, and the two are equal only when the numbers are equal. This inequality is the standard tool for minimising sums of the form x plus k by x, because the product x times k by x is fixed at k.
| Rule | Statement |
|---|---|
| AM of a and b | (a plus b) by 2 |
| GM of a and b | square root of ab |
| AM-GM | arithmetic mean at least geometric mean, for positive a and b |
| equality | only when a equals b |
Exam
In a geometric progression question, write the first term and the common ratio first, then translate each condition into an equation in those two unknowns. Two conditions fix both, and one condition typically leaves a quadratic in the ratio.