Sequences and Series

Numbers lined up in order form a sequence: 1, 3, 5, 7, … each number is a term. Number Patterns asked you to guess where such queues head; here we take the two most common squads all the way — arithmetic and geometric — and meet the big umbrella for adding: Σ.

Arithmetic sequences: steady steps

When every term exceeds the previous one by the same number, the line is an arithmetic sequence, and that number is the common difference d. Watch 3, 7, 11, 15, …:

Recursion view: next term = current term + 4;
Explicit view: the n-th term is $a_n = 3 + (n-1) \times 4$ , so the 10th term is $3 + 36 = 39$ .

Recursion is the drill command "stay glued to the person ahead"; the explicit formula is a roster — call a number, find the person directly, no counting from the front.

Gauss's pairing shortcut

What is 1 + 2 + … + 100? Legend says young Gauss answered 5050 before the teacher finished reading. His trick: pair the ends.

$1 + 2 + \cdots + 100 = (1+100) + (2+99) + \cdots = 101 \times 50 = 5050$

100 numbers make 50 pairs, each pair summing to 101. In general, the arithmetic series sums to

$S_n = \frac{n(a_1 + a_n)}{2}$

Check it on a small squad

The arithmetic sequence 3, 7, 11, …, 39 has 10 terms. Formula: $S_{10} = \frac{10 \times (3+39)}{2} = 210$ . Honest pairing: 3+39=42, 7+35=42, all five pairs are 42, $42 \times 5 = 210$ . Two roads, one answer.

Geometric sequences: faster and faster

When every term is multiplied by the same number, the line is a geometric sequence, and that number is the common ratio r. For 1, 2, 4, 8, … the explicit form is $a_n = 2^{n-1}$ , and the 20th term already reaches $2^{19} = 524288$ . The ever-growing curve of compound interest is exactly geometric growth.

Folding a sheet 42 times

A sheet 0.1 mm thick doubles in thickness with every fold: after 42 folds it is $0.1 \times 2^{42}$ mm — about $4.4 \times 10^{11}$ mm, some 440000 km, beyond the 380000 km to the Moon. No sheet can actually survive 42 folds, but the numbers truly dare to grow.

Reading the summation sign Σ

The capital Greek letter Σ (sigma) is shorthand for "add them all up":

$\sum_{k=1}^{100} k = 1 + 2 + \cdots + 100 = 5050$

Read it as "let k run from 1 to 100 and add each one". k is the summation index: the subscript is the start, the superscript the finish. One more read: $\sum_{k=1}^{5} k^2 = 1 + 4 + 9 + 16 + 25 = 55$ . And some sums never quite arrive: $1 + \frac{1}{2} + \frac{1}{4} + \cdots$ keeps approaching 2 yet always falls a little short — the story of infinite series continues in Limits.

Check yourself

Quick quiz

1. What is the 20th term of the arithmetic sequence 5, 9, 13, …?
2. What is 1 + 2 + 3 + … + 100?
3. What is the rule of the sequence 2, 6, 18, 54, …?