A sequence is an ordered list of numbers, often following a specific pattern or rule. Each number in the sequence is called a term.
Example: 2, 4, 6, 8, ... (Sequence of even numbers)
A series is the sum of the terms in a sequence.
Example: 2 + 4 + 6 + 8 + ... (Series corresponding to the sequence of even numbers)
We often talk about the nth partial sum (Sn) of a series, which is the sum of the first n terms of the corresponding sequence.
An arithmetic sequence is a sequence where the difference between consecutive terms is constant. This constant difference is called the common difference (d).
5, 8, 11, 14, ... (First term a = 5, common difference d = 3)an): an = a + (n-1)d
(where a is the first term, n is the term number)An arithmetic series is the sum of the terms of an arithmetic sequence.
Sn):
Sn = n/2 * (2a + (n-1)d) or equivalently Sn = n/2 * (a + an)A geometric sequence is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r).
3, 6, 12, 24, ... (First term a = 3, common ratio r = 2)an): an = a * r(n-1)
(where a is the first term, n is the term number)A geometric series is the sum of the terms of a geometric sequence.
Sn), provided r ≠ 1:
Sn = a * (1 - rn) / (1 - r)r = 1, the series is simply a + a + ... + a, so Sn = n * a.
An infinite geometric series is the sum of an infinite number of terms from a geometric sequence. This sum only makes sense (converges to a finite value) if the absolute value of the common ratio is less than 1 (|r| < 1).
|r| < 1, the series converges, and the infinite sum (S∞) is:
S∞ = a / (1 - r)|r| ≥ 1, the series diverges, meaning the sum does not approach a finite value (it goes to infinity or oscillates).| Series | First Term (a) | Common Ratio (r) | Infinite Sum (S∞ = a / (1 - r)) |
|---|---|---|---|
| 1 + 1/2 + 1/4 + 1/8 + ... | 1 | 1/2 | 1 / (1 - 1/2) = 2 |
| 1 + 1/3 + 1/9 + 1/27 + ... | 1 | 1/3 | 1 / (1 - 1/3) = 3/2 = 1.5 |
| 0.9 + 0.09 + 0.009 + ... | 0.9 | 0.1 | 0.9 / (1 - 0.1) = 0.9 / 0.9 = 1 |
| a - ar + ar2 - ar3 + ... (|r| < 1) | a | -r | a / (1 - (-r)) = a / (1 + r) |
The Fibonacci sequence is another famous sequence, but it's neither arithmetic nor geometric. It starts with 0 and 1, and each subsequent term is the sum of the two preceding ones.
Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, ...
Formula: Fn = Fn-1 + Fn-2, with F0 = 0 and F1 = 1.
This sequence appears surprisingly often in nature, art, and computer science.