Chapter 5: Perform sequence and series operations

Sequences and series are fundamental in discrete mathematics and have practical applications in ICT systems analysis, algorithm design, and data structuring. For ICT technicians in Kenya, understanding how to operate with sequences and series supports tasks such as estimating resource usage over time, analyzing network packet sequences, and calculating costs in staged infrastructure deployment. This chapter focuses on performing operations on sequences and series, emphasizing arithmetic sequences and series, which frequently appear in scheduling and budgeting scenarios within ICT projects.

5.1 Summation of sequence

A sequence is an ordered list of numbers, and summation is the process of adding the terms of a sequence to find a total. ICT technicians meet sequences whenever readings or counts are recorded in order, for example the sizes of successive data packets, the time between successive events or the number of print jobs handled each day. Adding the terms gives the total packet size, the total time or the total workload.

A sum is written with the letter S and the number of terms added. The sum of the first five terms of a sequence is:

S₅ = a₁ + a₂ + a₃ + a₄ + a₅

The same sum is written compactly with the Greek capital letter sigma (Σ), which means "add up": Σ aₖ for k = 1 to 5 adds the terms aₖ as k runs from 1 to 5.

Example: A technician records the number of print jobs handled by a workshop printer on five days: 12, 15, 18, 21, 24. The total is the sum of the five terms.

S₅ = 12 + 15 + 18 + 21 + 24
S₅ = 90

Answer: 90 print jobs

5.1.1 Key terms of sequences

A sequence is written as a₁, a₂, a₃, ..., aₙ, where the letter a stands for the terms and the small number is the position of the term in the list. The ordering matters: 2, 4, 6, 8 and 8, 6, 4, 2 are different sequences. The terms below describe a sequence and are needed before sums can be written or calculated.

5.1.1.1 Term

A term is one of the numbers in a sequence. The first term is written a₁, the second term a₂ and the nth term aₙ. Example: in the sequence 2, 4, 6, 8, the first term is a₁ = 2, the second term is a₂ = 4 and the fourth term is a₄ = 8. Summation adds terms, so identifying the first term and the last term is the first step in any sum.

5.1.1.2 Index

The index of a term is its position number in the sequence. It is written as a small subscript: in a₃ the index is 3, so a₃ is the third term. In mathematics the index of the first term is 1. Programming languages such as Python count from 0, so the term aₙ is stored at position n − 1 in a list.

Example: the list readings = [5, 8, 11, 14] holds a sequence in Python. The term a₁ = 5 is readings[0], and the term a₄ = 14 is readings[3].

5.1.1.3 General term (nth term)

The general term, or nth term, is a rule that gives any term of the sequence from its index n. It lets a technician find the 50th term without writing out the first 49. The general term is written aₙ = (a formula in n).

Example: aₙ = 3n + 2 gives a₁ = 3 × 1 + 2 = 5, a₂ = 8, a₃ = 11 and a₄ = 14, which is the sequence 5, 8, 11, 14, ... The 20th term is a₂₀ = 3 × 20 + 2 = 62.

When the difference between successive terms is a constant, called the common difference d, the general term is aₙ = a₁ + (n − 1) × d, where a₁ is the first term. This form is developed fully under General form of an arithmetic sequence; here it is used to find terms.

Example 1: Given the sequence 2, 4, 6, 8, find the first term, the common difference and the 5th term.

a₁ = 2
d = 4 − 2 = 2
aₙ = a₁ + (n − 1) × d
a₅ = 2 + (5 − 1) × 2
a₅ = 2 + 8
a₅ = 10

Answer: a₁ = 2, d = 2, a₅ = 10

Example 2: Find the 10th term of the sequence 5, 8, 11, 14, ...

a₁ = 5, d = 3, n = 10
a₁₀ = 5 + (10 − 1) × 3
a₁₀ = 5 + 27
a₁₀ = 32

Answer: 32

Example 3: Find the 15th term of the sequence 100, 95, 90, 85, ...

a₁ = 100, d = −5, n = 15
a₁₅ = 100 + (15 − 1) × (−5)
a₁₅ = 100 − 70
a₁₅ = 30

Answer: 30

5.1.1.4 Finite sequence

A finite sequence has a first term and a last term, so it has a definite number of terms n. The sequence 5, 8, 11, 14 is finite: it has n = 4 terms, with a₁ = 5 as the first term and a₄ = 14 as the last. Any sequence from a recorded set of readings is finite, because the recording stops. The sum of a finite sequence can always be found, by adding all its terms.

S₄ = 5 + 8 + 11 + 14
S₄ = 38

Example: a log server saves the size in kilobytes of four daily log files: 20, 25, 30, 35. This is a finite sequence with n = 4 terms, and the total size is 20 + 25 + 30 + 35 = 110 kilobytes.

5.1.1.5 Infinite sequence

An infinite sequence has a first term but no last term: it continues without end. It is written with three dots after the last term shown, for example 2, 4, 8, 16, ... An infinite sequence is described by its general term, which gives a term for every index n = 1, 2, 3, and so on.

Example: aₙ = 2ⁿ describes the infinite sequence 2, 4, 8, 16, 32, ... With n binary digits (bits) a computer can represent 2ⁿ different values, so the 8th term a₈ = 2⁸ = 256 is the number of values that 8 bits can represent. Terms can be calculated for as large an index as needed, and the sum of the first n terms can be found for any chosen n, but the terms can never all be listed.

Chapter Summary

This chapter covered the essential operations involving sequences and series, starting with the summation of sequences and introducing key terms that define sequence characteristics. It then explored arithmetic series, detailing how to calculate the arithmetic sum and presenting the general form of an arithmetic sequence. The concept of arithmetic progression was explained to show how terms increase by a constant difference. The chapter continued with geometric series, describing the general form of a geometric sequence where terms change by a constant ratio. Finally, geometric progression was examined to illustrate how terms multiply by a common ratio to form the sequence. Throughout, the focus remained on understanding and performing calculations with these sequences and series in discrete mathematics.