By the end of this chapter, you will be able to:
These skills will help you solve real-world problems where patterns and progressions are key, making your work in science and technology more precise and effective.
Applied Biology professionals often encounter data patterns and trends that can be analyzed using sequences and series. Understanding arithmetic sequences and series is crucial for modelling biological growth, decay, and periodic measurements such as population counts, enzyme activity rates, or nutrient concentrations over time. This chapter focuses on arithmetic sequences and series, providing tools to calculate terms and sums, essential for interpreting and forecasting biological data in Kenya's agricultural, environmental, and health sectors.
Arithmetic sequences appear frequently in biological data where changes occur by a constant amount over equal intervals, such as the steady increase of plant height or linear decline in population due to harvesting. Mastery of arithmetic sequences allows Applied Biology practitioners to predict future values and analyze trends effectively.
The arithmetic mean of two numbers in a sequence is the value that lies exactly halfway between them, assuming the sequence increases or decreases by a constant difference. For any two terms \(a\) and \(b\), the arithmetic mean \(M\) is calculated as:
$$ M = \frac{a + b}{2} $$
Example 1: In a plant growth experiment, the height after day 4 is 24 cm and after day 6 is 32 cm. Find the arithmetic mean height for day 5.
Given: \(a = 24 \text{ cm}\), \(b = 32 \text{ cm}\)
$$ M = \frac{24 + 32}{2} $$
$$ M = \frac{56}{2} $$
$$ M = 28 \text{ cm} $$
Answer: 28 cm
Example 2: The concentration of a nutrient solution is 10 mg/L on Monday and 16 mg/L on Wednesday. Find the arithmetic mean concentration on Tuesday.
Given: \(a = 10 \text{ mg/L}\), \(b = 16 \text{ mg/L}\)
$$ M = \frac{10 + 16}{2} $$
$$ M = \frac{26}{2} $$
$$ M = 13 \text{ mg/L} $$
Answer: 13 mg/L
Example 3: A population of fish in a pond is 150 after 2 weeks and 210 after 6 weeks. Find the arithmetic mean population after 4 weeks.
Given: \(a = 150\), \(b = 210\)
$$ M = \frac{150 + 210}{2} $$
$$ M = \frac{360}{2} $$
$$ M = 180 $$
Answer: 180 fish
The nth term of an arithmetic sequence is the value of the term at position \(n\), where the difference between consecutive terms is constant. The formula for the nth term \(a_n\) is:
$$ a_n = a_1 + (n - 1)d $$
where \(a_1\) is the first term and \(d\) is the common difference.
Example 1: In a bacterial culture, the population increases by 500 every hour. If the initial population is 2000, find the population after 5 hours.
Given: \(a_1 = 2000\), \(d = 500\), \(n = 5\)
$$ a_n = 2000 + (5 - 1) \times 500 $$
$$ a_n = 2000 + 4 \times 500 $$
$$ a_n = 2000 + 2000 $$
$$ a_n = 4000 $$
Answer: 4000 bacteria
Example 2: A farmer plants 100 trees in the first row and increases the number by 20 trees in each subsequent row. How many trees are in the 10th row?
Given: \(a_1 = 100\), \(d = 20\), \(n = 10\)
$$ a_n = 100 + (10 - 1) \times 20 $$
$$ a_n = 100 + 9 \times 20 $$
$$ a_n = 100 + 180 $$
$$ a_n = 280 $$
Answer: 280 trees
Example 3: A researcher measures enzyme activity starting at 50 units and decreases by 3 units each day. Find the enzyme activity on day 15.
Given: \(a_1 = 50\), \(d = -3\), \(n = 15\)
$$ a_n = 50 + (15 - 1) \times (-3) $$
$$ a_n = 50 + 14 \times (-3) $$
$$ a_n = 50 - 42 $$
$$ a_n = 8 $$
Answer: 8 units
Example 4: A water sample’s pH increases by 0.2 units every hour starting from 6.5. What is the pH after 12 hours?
Given: \(a_1 = 6.5\), \(d = 0.2\), \(n = 12\)
$$ a_n = 6.5 + (12 - 1) \times 0.2 $$
$$ a_n = 6.5 + 11 \times 0.2 $$
$$ a_n = 6.5 + 2.2 $$
$$ a_n = 8.7 $$
Answer: 8.7 pH units
Example 5: The depth of water in a dam decreases by 0.15 m per day from an initial depth of 8 m. Find the depth after 30 days.
Given: \(a_1 = 8\), \(d = -0.15\), \(n = 30\)
$$ a_n = 8 + (30 - 1) \times (-0.15) $$
$$ a_n = 8 + 29 \times (-0.15) $$
$$ a_n = 8 - 4.35 $$
$$ a_n = 3.65 $$
Answer: 3.65 m
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Create a free accountThis chapter covered the fundamental concepts of sequences and series, beginning with arithmetic sequences where the difference between consecutive terms is constant. The arithmetic mean was introduced as the value that lies between two numbers in an arithmetic sequence, and the formula for the nth term of an arithmetic sequence was derived and applied. The chapter then explored the sum of terms in an arithmetic series, also known as arithmetic progression, providing methods to calculate the total of a specified number of terms. Moving to geometric sequences, the chapter explained sequences where each term is obtained by multiplying the previous term by a fixed ratio. Finite geometric sequences were examined along with the concept of geometric means and the formula for finding the nth term. Finally, the chapter detailed how to find the sum of both finite and infinite geometric sequences, emphasizing the conditions under which the infinite sum converges. Throughout, the focus remained on practical calculation techniques relevant to science and engineering contexts.
In a population of bacteria, the number of bacteria increases by 20 every hour starting from 100 bacteria. Find the number of bacteria after 10 hours. (2 marks)
Calculate the arithmetic mean of the sequence: 12, 18, 24, 30, 36. (2 marks)
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