Lesson 7: Summation and Series

Sequences

A sequence is an ordered list of numbers. It may be finite or infinite.

Notation: a₁, a₂, a₃, ..., aₙ

Example: {1, 2, 3, 4, ...} where aₙ = n.

Arithmetic Progression

An arithmetic progression is a sequence where consecutive terms have a constant difference called the common difference (d).

d = a₂ − a₁

Recursive formula (nth term): aₙ = a₁ + d(n − 1)

Example: {40, 37, 34, 31, ...} d = 37 − 40 = −3 a₄ = 40 + (−3)(3) = 31

Geometric Progression

A geometric progression multiplies each term by a constant called the common ratio (r).

r = a₂ / a₁

nth term formula: aₙ = a · r^(n−1)

Example: a = 15, r = 3, find the 5th term. a₅ = 15 × 3⁴ = 15 × 81 = 1215

Fibonacci Sequence

A sequence where each term is the sum of the two preceding terms:

xₙ = xₙ₋₁ + xₙ₋₂

Starting from: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

Example: x₈ = x₇ + x₆ = 13 + 8 = 21

Summation (Sigma Notation)

Summation (∑) is a compact way to write the sum of terms in a sequence.

∑(i=k to n) aᵢ = a_k + a_(k+1) + ... + a_n

Where i = index, k = lower limit, n = upper limit.

Example: ∑(j=3 to 5) (3j − 1) = (3(3)−1) + (3(4)−1) + (3(5)−1) = 8 + 11 + 14 = 33

Useful standard results:

• ∑ 1 (from i=1 to n) = n
• ∑ i (from i=1 to n) = n(n+1)/2

Arithmetic Series

The sum of the first n terms of an arithmetic progression:

S = n × (a₁ + aₙ) / 2

Example: Sum of 2 + 4 + 6 + ... + 100 (n = 50): S = 50 × (2 + 100) / 2 = 50 × 51 = 2550

Geometric Series

The sum of the first n terms of a geometric progression:

S = a × (rⁿ − 1) / (r − 1)