Sequences and Series
Arithmetic and geometric sequences, their sums, sigma notation, and how series connect to exponential growth and infinite sums.
Sequences
A sequence is an ordered list of numbers following a pattern. Each number in the list is a term.
2, 5, 8, 11, 14, ... (arithmetic — add 3 each time)
3, 6, 12, 24, 48, ... (geometric — multiply by 2 each time)
1, 1, 2, 3, 5, 8, ... (Fibonacci — each term is sum of previous two)
Notation: the nth term is written aₙ. The first term is a₁ (or sometimes a₀).
Arithmetic Sequences
Each term differs from the previous by a constant — the common difference d.
aₙ = a₁ + (n − 1)d
Sequence: 3, 7, 11, 15, ...
a₁ = 3, d = 4
a₁₀ = 3 + 9 × 4 = 39
a₁₀₀ = 3 + 99 × 4 = 399
Finding d and a₁
If you know two terms:
a₅ = 17, a₁₀ = 37
d = (37 − 17)/(10 − 5) = 4
a₁ = 17 − 4 × 4 = 1
Linear connection
Arithmetic sequences are linear functions of n. Plot aₙ vs n and you get a straight line with slope d. This is why arithmetic and linear functions are intimately related.
Geometric Sequences
Each term is multiplied by a constant — the common ratio r.
aₙ = a₁ × rⁿ⁻¹
Sequence: 2, 6, 18, 54, ...
a₁ = 2, r = 3
a₅ = 2 × 3⁴ = 162
a₁₀ = 2 × 3⁹ = 39,366
Finding r
r = aₙ₊₁ / aₙ (any consecutive pair)
Exponential connection
Geometric sequences are exponential functions of n. Plot aₙ vs n and you get an exponential curve. Population growth, compound interest, viral spread — all geometric sequences in disguise.
Series — Summing a Sequence
A series is the sum of terms of a sequence. Denoted Sₙ for the sum of the first n terms.
Arithmetic Series
Sum of the first n terms of an arithmetic sequence:
Sₙ = n/2 × (a₁ + aₙ) (average of first and last, times n)
Sₙ = n/2 × (2a₁ + (n−1)d) (expanded form)
Gauss’s insight: sum 1 + 2 + 3 + … + 100. Pair terms from both ends: (1+100) + (2+99) + … = 101 × 50 = 5050. This is the formula with a₁ = 1, aₙ = 100, n = 100.
Sum of 1 + 2 + ... + n = n(n+1)/2
This appears constantly — in combinatorics, algorithm analysis, and physics.
Sequence: 5, 9, 13, ..., 41 (d=4)
n = (41−5)/4 + 1 = 10
S₁₀ = 10/2 × (5 + 41) = 230
Geometric Series
Sum of the first n terms of a geometric sequence:
Sₙ = a₁(1 − rⁿ) / (1 − r) (r ≠ 1)
Derivation: Write S = a + ar + ar² + … + arⁿ⁻¹. Multiply by r: rS = ar + ar² + … + arⁿ. Subtract: S − rS = a − arⁿ. Factor: S(1−r) = a(1−rⁿ). Done.
Sum of 3 + 6 + 12 + 24 + ... (8 terms), a₁=3, r=2
S₈ = 3(1 − 2⁸)/(1 − 2) = 3(1 − 256)/(−1) = 3 × 255 = 765
Infinite Geometric Series
If |r| < 1, the terms shrink and the series converges to a finite sum:
S∞ = a₁ / (1 − r) (only when |r| < 1)
1 + 1/2 + 1/4 + 1/8 + ... = 1/(1 − 1/2) = 2
0.999... = 9/10 + 9/100 + ... = (9/10)/(1 − 1/10) = 1
The last one is worth pausing on: 0.999… = 1 exactly, not approximately. The infinite series converges to 1.
If |r| ≥ 1, the series diverges — the sum grows without bound (or oscillates).
Sigma Notation
Sigma (Σ) notation is shorthand for a sum:
Σᵢ₌₁ⁿ f(i) = f(1) + f(2) + f(3) + ... + f(n)
- Bottom: starting index (i = 1)
- Top: ending index (n)
- Right: the term formula
Σᵢ₌₁⁵ i = 1 + 2 + 3 + 4 + 5 = 15
Σᵢ₌₁⁴ 2ⁱ = 2 + 4 + 8 + 16 = 30
Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6
Key sigma identities
Σ c = cn (sum of constant)
Σ i = n(n+1)/2 (sum of first n integers)
Σ i² = n(n+1)(2n+1)/6 (sum of squares)
Σ i³ = [n(n+1)/2]² (sum of cubes = square of sum of integers)
These identities come up in algorithm complexity analysis — summing loop iterations.
Connecting to Exponential Growth
A geometric sequence with r > 1 grows exponentially. The connection is exact:
aₙ = a₁ · rⁿ⁻¹ = a₁ · eⁿ⁻¹·ln(r)
Every geometric sequence is an exponential function evaluated at integers. When you go from discrete (sequence) to continuous (function), the geometric sequence becomes the exponential function.
This is why compound interest (geometric series) and continuous compounding (exponential function) give related but slightly different answers — one is discrete, the other continuous.
Fibonacci and Recurrence Relations
Not all sequences are arithmetic or geometric. A recurrence relation defines each term from previous ones:
Fibonacci: F₁ = 1, F₂ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
The ratio of consecutive Fibonacci terms converges to the golden ratio φ ≈ 1.618.
Recurrence relations appear in algorithm analysis (how fast does this recursive function run?), population models, and signal processing.