Sequences and series
A sequence is an ordered list of numbers a₁, a₂, a₃, … where each term is linked to its position. A series is the sum of the terms of a sequence: a₁ + a₂ + a₃ + …
Sequences can be described by a formula for the nth term (for example aₙ = n² + 1) or by a rule relating each term to earlier ones. The Fibonacci sequence uses aₙ = aₙ₋₁ + aₙ₋₂.
Arithmetic progression (AP)
Each term differs from the previous one by a common difference d.
aₙ = a + (n − 1)d
Sₙ = n/2 [2a + (n − 1)d] = n/2 (a + l)
where a is the first term and l the last.
Useful facts:
Three numbers in AP can be written as a − d, a, a + d. This makes many problems easier.
If each term of an AP is increased, decreased, multiplied or divided (by a non-zero number) by the same constant, the result is still an AP.
Arithmetic mean
The arithmetic mean (AM) of a and b is A = (a + b)/2, so that a, A, b form an AP.
To insert n arithmetic means between a and b, use the common difference d = (b − a)/(n + 1).
Geometric progression (GP)
Each term is obtained by multiplying the previous one by a common ratio r (r ≠ 0).
aₙ = a rⁿ⁻¹
Sₙ = a(rⁿ − 1)/(r − 1) for r ≠ 1, and Sₙ = na if r = 1
Example. Find the sum of the first 6 terms of 3, 6, 12, …
a = 3, r = 2: S₆ = 3(2⁶ − 1)/(2 − 1) = 3 × 63 = 189.
Three numbers in GP can be written as a/r, a, ar.
Sum of an infinite GP
If |r| < 1, then rⁿ → 0 as n grows, and the infinite sum converges:
S∞ = a / (1 − r)
Example. 1 + 1/2 + 1/4 + 1/8 + … = 1 / (1 − ½) = 2.
Example. Write 0.333… as a fraction. It is 3/10 + 3/100 + … with a = 3/10 and r = 1/10, so S∞ = (3/10)/(9/10) = 1/3.
If |r| ≥ 1, the infinite GP has no finite sum.
Geometric mean and AM ≥ GM
The geometric mean (GM) of two positive numbers a and b is G = √(ab), so that a, G, b form a GP.
For any two positive numbers:
AM ≥ GM, that is, (a + b)/2 ≥ √(ab), with equality only when a = b
Proof. (a + b)/2 − √(ab) = (√a − √b)² / 2, which is never negative.
This inequality is very useful in optimisation. For example, for positive x, x + 1/x ≥ 2, with the minimum value 2 at x = 1.
Special sums
Sum | Formula |
|---|---|
1 + 2 + … + n | n(n + 1)/2 |
1² + 2² + … + n² | n(n + 1)(2n + 1)/6 |
1³ + 2³ + … + n³ | [n(n + 1)/2]² |
Example. 1² + 2² + … + 10² = 10 × 11 × 21 / 6 = 385.
AP or GP?
Constant difference between consecutive terms: AP.
Constant ratio between consecutive terms: GP.
The sequence 2, 6, 18, 54 has a constant ratio of 3, so it is a GP, even though the differences (4, 12, 36) grow.