Algebra › Sequences & series
Sequences & series
A sequence is a list of numbers in order. A series is what you get when you add them up. Both live or die by the rule that connects them.
Build a sequence
Pick a rule, set the start, watch the pattern unfold.
Rule
aₙ = a + (n−1)d
Each term is +3 from the one before. This is an arithmetic sequence with common difference d = 3.
Two famous families
Arithmetic sequences
Each term is the previous one plus a fixed amount called the common difference d. Example with d = 3: 2, 5, 8, 11, 14…
The n-th term is aₙ = a₁ + (n − 1) · d. The sum of the first n terms is Sₙ = n(a₁ + aₙ) / 2.
Geometric sequences
Each term is the previous one times a fixed amount called the common ratio r. Example with r = 2: 3, 6, 12, 24, 48…
The n-th term is aₙ = a₁ · r^(n − 1). The sum of the first n terms is Sₙ = a₁ (1 − rⁿ) / (1 − r).
Series — when the sum settles down
|r| < 1, the infinite sum has a finite value: S = a₁ / (1 − r). So 1 + ½ + ¼ + ⅛ + … sums to 2.Other patterns to spot
- Square numbers — 1, 4, 9, 16, 25 (each one is n²).
- Triangular numbers — 1, 3, 6, 10, 15 (each one is n(n+1)/2).
- Fibonacci — 1, 1, 2, 3, 5, 8 (each is the sum of the previous two).
Finding the rule
Look at the differences between terms. If the differences are constant, the sequence is arithmetic. If they double or triple, it's geometric. If the differences themselves form a sequence, you might be looking at squares or cubes.
Quick check
- Find the 10th term of 5, 8, 11, 14, …
- Find the 6th term of 2, 6, 18, 54, …
- Sum the first 100 positive integers.
Answers: 32; 486; 5050.
What's the next term in 2, 6, 18, 54, …?
The sum 1 + ½ + ¼ + ⅛ + … goes on forever. What does it add to?