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Sequences are fundamental in discrete mathematics, providing a structured way to organize data and analyze patterns in various professional fields. From tracking patient visits in a county referral hospital to monitoring monthly sales in a retail business or scheduling activities in a cooperative, sequences help professionals model and predict outcomes. Understanding the terminology and properties of sequences is essential for accurate data interpretation and decision-making. This chapter introduces key terms related to sequences, forming the basis for subsequent operations on sequences and series.
Key terms define the language of sequences, enabling clear communication and precise calculations in professional contexts such as banking transaction records, agricultural yield tracking, or inventory control in hotels. Mastery of these terms allows for effective manipulation of sequences in various applications.
A term in a sequence is an individual element or member of the sequence, identified by its position. Terms represent specific data points, such as daily rainfall amounts recorded by a county government meteorological office or weekly production units in a dairy farm.
The general formula for the \(n\)th term of a sequence is often expressed as \(a_n\), where \(n\) is the term's position.
$$a_n = \text{value of the term at position } n$$
Example 1: A retail store records the number of customers visiting daily as the sequence 10, 15, 20, 25, ... Find the 4th term.
Given: Sequence \(a_1 = 10\), \(a_2 = 15\), \(a_3 = 20\), \(a_4 = ?\)
This is an arithmetic sequence increasing by 5 each day.
$$a_n = a_1 + (n-1)d$$
$$a_4 = 10 + (4-1) \times 5$$
$$a_4 = 10 + 3 \times 5$$
$$a_4 = 10 + 15$$
Answer: 25 customers
Example 2: In a cooperative, the number of bags of maize harvested each month forms the sequence: 100, 90, 80, 70, ... Find the 5th term.
Given: \(a_1 = 100\), common difference \(d = -10\), find \(a_5\)
$$a_n = a_1 + (n-1)d$$
$$a_5 = 100 + (5-1)(-10)$$
$$a_5 = 100 - 40$$
$$a_5 = 60$$
Answer: 60 bags
Example 3: A hotel records the number of rooms booked each day in a week as \(a_1 = 30\), \(a_2 = 35\), \(a_3 = 40\), and so on, increasing by 5 daily. Find the 10th term.
Given: \(a_1 = 30\), \(d=5\), find \(a_{10}\)
$$a_n = a_1 + (n-1)d$$
$$a_{10} = 30 + (10-1) \times 5$$
$$a_{10} = 30 + 45$$
$$a_{10} = 75$$
Answer: 75 rooms
The index of a term denotes the position of that term within a sequence. It is commonly represented by the variable \(n\), a positive integer starting from 1 or 0 depending on the context. For example, in a bank’s transaction log, the index may represent the transaction number.
The index helps identify and access specific terms for analysis.
$$n = \text{position of the term in the sequence}$$
Example 1: Given the sequence 5, 10, 15, 20, ..., find the index of the term 35.
Given: \(a_n = 5n\), find \(n\) when \(a_n = 35\)
$$a_n = 5n$$
$$35 = 5n$$
$$n = \frac{35}{5}$$
$$n = 7$$
Answer: The term 35 is at position 7
Example 2: A sequence represents daily patient visits at a hospital: 12, 15, 18, 21, ... Find the index of the term 33.
Given: \(a_n = 12 + (n-1) \times 3\), find \(n\) when \(a_n = 33\)
$$33 = 12 + 3(n-1)$$
$$33 - 12 = 3(n-1)$$
$$21 = 3n - 3$$
$$21 + 3 = 3n$$
$$24 = 3n$$
$$n = \frac{24}{3}$$
$$n = 8$$
Answer: The term 33 is at position 8
Example 3: In a sequence defined by \(a_n = 2n^2\), find the index of the term 98.
Given: \(a_n = 2n^2\), find \(n\) when \(a_n=98\)
$$98 = 2n^2$$
$$n^2 = \frac{98}{2}$$
$$n^2 = 49$$
$$n = \sqrt{49}$$
$$n = 7$$
Answer: The term 98 is at position 7
The general term or nth term of a sequence is a formula that defines the value of any term based on its position \(n\). It allows generation of terms without listing all preceding terms. For example, the general term can model monthly revenue growth in a SACCO or daily water consumption in a county office.
The general term is denoted as \(a_n\).
$$a_n = \text{expression involving } n$$
Example 1: Find the general term of the arithmetic sequence 7, 10, 13, 16, ...
Given: First term \(a_1 = 7\), common difference \(d = 3\)
$$a_n = a_1 + (n-1)d$$
$$a_n = 7 + (n-1) \times 3$$
$$a_n = 7 + 3n - 3$$
$$a_n = 3n + 4$$
Answer: \(a_n = 3n + 4\)
Example 2: Determine the general term of the sequence 2, 4, 8, 16, 32, ...
This is a geometric sequence with first term \(a_1 = 2\) and common ratio \(r = 2\).
$$a_n = a_1 \times r^{n-1}$$
$$a_n = 2 \times 2^{n-1}$$
$$a_n = 2^{n}$$
Answer: \(a_n = 2^{n}\)
Example 3: Find the general term of the sequence 1, 4, 9, 16, 25, ...
These are perfect squares, so:
$$a_n = n^2$$
Answer: \(a_n = n^{2}\)
A finite sequence is a sequence that contains a limited number of terms. This type of sequence is common in practical scenarios such as a hotel’s guest list for a week or a bank’s monthly transaction record. Finite sequences can be explicitly listed or described by a formula with specified term limits.
The number of terms \(N\) is finite.
$$\{a_1, a_2, \dots, a_N\}$$
Example 1: Write the finite sequence of daily sales in a shop for 5 days given by the arithmetic sequence \(a_n = 5n + 10\).
Given: \(n = 1, 2, 3, 4, 5\)
Calculate terms:
$$a_1 = 5(1) + 10 = 15$$
$$a_2 = 5(2) + 10 = 20$$
$$a_3 = 5(3) + 10 = 25$$
$$a_4 = 5(4) + 10 = 30$$
$$a_5 = 5(5) + 10 = 35$$
Answer: The finite sequence is {15, 20, 25, 30, 35}
Example 2: Find the sum of the first 6 terms of the sequence \(a_n = 3n^2\).
Given: \(n = 1 \text{ to } 6\)
Calculate terms:
$$a_1 = 3(1)^2 = 3$$
$$a_2 = 3(2)^2 = 12$$
$$a_3 = 3(3)^2 = 27$$
$$a_4 = 3(4)^2 = 48$$
$$a_5 = 3(5)^2 = 75$$
$$a_6 = 3(6)^2 = 108$$
Sum:
$$S_6 = 3 + 12 + 27 + 48 + 75 + 108 = 273$$
Answer: The sum of the first 6 terms is 273
Example 3: A cooperative distributes dividends to members for 4 years following the sequence \(a_n = 1000 + 200n\). List the sequence and find the total amount paid over the 4 years.
Calculate terms:
$$a_1 = 1000 + 200(1) = 1200$$
$$a_2 = 1000 + 200(2) = 1400$$
$$a_3 = 1000 + 200(3) = 1600$$
$$a_4 = 1000 + 200(4) = 1800$$
Total:
$$S_4 = 1200 + 1400 + 1600 + 1800 = 6000$$
Answer: Sequence: {1200, 1400, 1600, 1800}, Total = 6000
An infinite sequence is a sequence that continues indefinitely without end. These sequences often model ongoing processes such as continuous data streams in ICT systems or ongoing monthly water usage in a county government. Infinite sequences are generally expressed using a formula for the \(n\)th term without an upper limit on \(n\).
Infinite sequences are denoted as:
$$\{a_1, a_2, a_3, \dots\}$$
with \(n \to \infty\).
Example 1: Consider the infinite geometric sequence \(a_n = 5 \times (0.5)^{n-1}\). Find the 10th term.
Given: \(a_n = 5 \times (0.5)^{n-1}\), \(n=10\)
$$a_{10} = 5 \times (0.5)^{10 - 1}$$
$$a_{10} = 5 \times (0.5)^9$$
$$a_{10} = 5 \times 0.001953125$$
$$a_{10} = 0.009765625$$
Answer: The 10th term is 0.009765625
Example 2: Find the 15th term of the infinite sequence defined by \(a_n = 3n + 2\).
Given: \(a_n = 3n + 2\), \(n=15\)
$$a_{15} = 3 \times 15 + 2$$
$$a_{15} = 45 + 2$$
$$a_{15} = 47$$
Answer: The 15th term is 47
Example 3: Determine the 20th term of the sequence \(a_n = \frac{10}{n}\).
Given: \(a_n = \frac{10}{n}\), \(n=20\)
$$a_{20} = \frac{10}{20}$$
$$a_{20} = 0.5$$
Answer: The 20th term is 0.5
Given the arithmetic sequence \(a_n = 4n + 1\), find the 12th term. (3 marks)
A sequence is defined by \(a_n = 50 - 3n\). Find the index \(n\) when \(a_n = 23\). (4 marks)
The finite sequence of monthly salaries for 6 months is given by \(a_n = 20000 + 1500n\). Calculate the total salary paid over the 6 months. (5 marks)
Find the 8th term of the infinite geometric sequence \(a_n = 3 \times (0.8)^{n-1}\). (4 marks)
Determine the general term of the sequence: 10, 20, 30, 40, ... (3 marks)
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Create a free accountThis chapter introduced the fundamental concepts of sequences, beginning with key terms such as term, index, and the general term or nth term. It distinguished between finite sequences, which have a limited number of terms, and infinite sequences that continue indefinitely. The chapter then explained the summation of sequences, focusing on the arithmetic sum as a way to add terms in a sequence. It detailed the general form of an arithmetic sequence, where each term increases by a constant difference, and provided the formula for finding the sum of the first \(n\) terms. The chapter concluded with geometric series, describing the general form of a geometric sequence in which each term is multiplied by a constant ratio. Understanding these concepts equips students to perform operations on sequences and series effectively in various mathematical and engineering contexts.
A cooperative society records the number of new members joining each month as a sequence: 3, 6, 9, 12, ... Find the 15th term of this sequence. (2 marks)
A county hospital monitors daily patient admissions forming a sequence: 50, 53, 56, 59, ... Calculate the sum of admissions for the first 20 days. (3 marks)
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