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Apply sequences and series is a fundamental mathematical topic in science laboratory technology, helping professionals analyze patterns and predict values in data sets such as chemical concentrations, reaction rates, and measurement readings. Mastery of arithmetic sequences and series enables accurate data interpretation and efficient calculation of cumulative effects in experiments. This chapter focuses on arithmetic sequences, arithmetic means, nth terms, and the sum of arithmetic series, providing practical calculation skills essential for laboratory analysis.
Arithmetic sequences appear frequently in laboratory measurements where quantities increase or decrease by a constant amount, for example, temperature increments in a controlled experiment or periodic sampling intervals.
The arithmetic mean in a sequence is the average of two terms equidistant from the beginning and end, representing the central tendency of the data. It is calculated using the relation:
$$ A.M. = \frac{a + l}{2} $$
where \(a\) is the first term and \(l\) is the last term of the sequence.
Example 1: A laboratory technician records readings of a solution’s pH at equal time intervals, obtaining a first value of 4 and a last value of 10. Find the arithmetic mean of these readings.
Given: \(a = 4\), \(l = 10\)
$$ A.M. = \frac{a + l}{2} $$
$$ A.M. = \frac{4 + 10}{2} $$
$$ A.M. = \frac{14}{2} $$
Answer: 7
Example 2: In an experiment, the first recorded temperature is 20°C and the last is 50°C. Calculate the arithmetic mean temperature.
Given: \(a = 20\), \(l = 50\)
$$ A.M. = \frac{20 + 50}{2} $$
$$ A.M. = \frac{70}{2} $$
Answer: 35°C
Example 3: A series of absorbance readings start at 0.12 and end at 0.48. Find the arithmetic mean absorbance.
Given: \(a = 0.12\), \(l = 0.48\)
$$ A.M. = \frac{0.12 + 0.48}{2} $$
$$ A.M. = \frac{0.60}{2} $$
Answer: 0.30
Example 4: A set of concentration measurements in a titration starts at 0.5 M and ends at 2.5 M. Find the arithmetic mean concentration.
Given: \(a = 0.5\), \(l = 2.5\)
$$ A.M. = \frac{0.5 + 2.5}{2} $$
$$ A.M. = \frac{3.0}{2} $$
Answer: 1.5 M
Example 5: The initial and final weights of a sample are 150 g and 450 g respectively in a drying process. Calculate the arithmetic mean weight.
Given: \(a = 150\), \(l = 450\)
$$ A.M. = \frac{150 + 450}{2} $$
$$ A.M. = \frac{600}{2} $$
Answer: 300 g
The nth term defines the value of a term at position \(n\) in an arithmetic sequence where each term increases by a constant difference \(d\). The formula is:
$$ a_n = a + (n - 1)d $$
where \(a\) is the first term, \(d\) is the common difference, and \(n\) is the term number.
Example 1: The initial concentration of a reagent is 2 mol/L, increasing by 0.5 mol/L every hour. Find the concentration after 5 hours.
Given: \(a = 2\), \(d = 0.5\), \(n = 5\)
$$ a_n = a + (n - 1)d $$
$$ a_5 = 2 + (5 - 1) \times 0.5 $$
$$ a_5 = 2 + 4 \times 0.5 $$
$$ a_5 = 2 + 2 $$
Answer: 4 mol/L
Example 2: A laboratory scale records weights starting at 100 g, increasing by 15 g each measurement. Find the 8th measurement.
Given: \(a = 100\), \(d = 15\), \(n = 8\)
$$ a_8 = 100 + (8 - 1) \times 15 $$
$$ a_8 = 100 + 7 \times 15 $$
$$ a_8 = 100 + 105 $$
Answer: 205 g
Example 3: The temperature of a solution starts at 25°C and decreases by 3°C per hour. Find the temperature after 6 hours.
Given: \(a = 25\), \(d = -3\), \(n = 6\)
$$ a_6 = 25 + (6 - 1) \times (-3) $$
$$ a_6 = 25 + 5 \times (-3) $$
$$ a_6 = 25 - 15 $$
Answer: 10°C
Example 4: A sample’s pH starts at 7 and increases by 0.2 per measurement. Find the pH at the 12th measurement.
Given: \(a = 7\), \(d = 0.2\), \(n = 12\)
$$ a_{12} = 7 + (12 - 1) \times 0.2 $$
$$ a_{12} = 7 + 11 \times 0.2 $$
$$ a_{12} = 7 + 2.2 $$
Answer: 9.2
Example 5: In a titration, the volume of titrant added starts at 5 mL and is increased by 2 mL every step. Calculate the volume at the 20th step.
Given: \(a = 5\), \(d = 2\), \(n = 20\)
$$ a_{20} = 5 + (20 - 1) \times 2 $$
$$ a_{20} = 5 + 19 \times 2 $$
$$ a_{20} = 5 + 38 $$
Answer: 43 mL
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Create a free accountThis chapter covered the fundamentals of sequences and series, beginning with arithmetic sequences where each term increases by a constant difference. It introduced the concept of the arithmetic mean and demonstrated how to find the nth term of an arithmetic sequence. The chapter then explained how to calculate the sum of terms in an arithmetic series, also known as arithmetic progression. Moving to geometric sequences, it described sequences where each term is multiplied by a constant ratio and detailed the properties of finite geometric sequences. The concept of geometric means was explored along with methods to determine the nth term of a geometric sequence. Finally, the chapter addressed the sums of both finite and infinite geometric series, highlighting the conditions under which these sums converge. Throughout, practical calculations and formula applications were emphasized to build a strong foundation in handling sequences and series in scientific contexts.
A laboratory technician records a sequence of temperature readings increasing by 2°C each hour, starting from 20°C. What is the 10th temperature reading? (2 marks)
Calculate the arithmetic mean of the first 15 measurements in an arithmetic sequence where the first term is 5 and the common difference is 3. (2 marks)
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