Lecture Notes: Sequences, Series, and Financial Applications (Competencies)
Page 1 - Competencies (early portion):
Computing Salary, Wage, and Commission: Understanding fixed annual payments (salary, paid regularly regardless of hours worked), hourly rates (wage, dependent on hours worked), and earnings based on sales percentage (commission, often used in sales roles).
Computing Gross and Net Pay: Gross pay is total earnings before any deductions are made; net pay is the actual amount received by an individual after taxes (e.g., income tax, social security), insurance premiums (e.g., health, life), and other deductions (e.g., retirement contributions) have been subtracted.
Applying Percentages in Various Contexts: Calculating practical applications such as retail discounts, loan or investment interest rates, sales tax amounts on purchases, and determining percentage changes in quantities or values.
Patterns and Sequences (intro):
Identify and Describe Patterns: Recognizing repeating elements (e.g., A, B, A, B), increasing/decreasing progressions (e.g., 1, 3, 5, 7 where 2 is added each time), or visual arrangements which follow a predictable rule.
Describe and Classify Sequence: An ordered list of numbers (terms), where each number occupies a specific position. Examples include 1, 2, 3, 4 (an arithmetic sequence) or 2, 4, 8, 16 (a geometric sequence).
Determine Fibonacci Sequence: A numerical sequence where each number (starting from the third) is the sum of the two preceding ones. It typically begins with 0 and 1 (i.e., 0, 1, 1, 2, 3, 5, 8, \ldots) or 1 and 1 (i.e., 1, 1, 2, 3, 5, 8, \ldots).
Describe the rule for finding the next term in a sequence: Identifying the specific mathematical operation (addition, subtraction, multiplication, division, exponentiation, or a combination thereof) that connects consecutive terms, allowing for the prediction of future terms.
Describe a series: The sum of the terms of a sequence, representing the total accumulation of the values. For example, the series for the sequence 1, 2, 3 is 1 + 2 + 3 = 6.
Identify sequence and series: Crucially, a sequence is a list of elements, whereas a series is the sum of the terms within that list.
Explain the difference between a sequence and a series: A sequence is simply an ordered arrangement or list of elements (terms), while a series is the result obtained by adding all the terms in a sequence together.
Write a given series in sigma notation and vice-versa: Expressing a sum compactly using the summation symbol \sum (sigma), which specifies the general term, the starting point (lower limit), and the ending point (upper limit) of the summation. For example, the sum 1+2+3+4 can be written as \sum_{k=1}^4 k.
Arithmetic Sequence (intro):
Illustrate sequence and series: For example, 3, 6, 9, 12 is an arithmetic sequence (common difference is 3); 3 + 6 + 9 + 12 = 30 is an arithmetic series.
Define arithmetic sequence and its key components: A sequence in which the difference between consecutive terms is constant. This constant difference is known as the common difference (d), and the first term is typically denoted a_1 (or a_0).
Key formula: nth term of an arithmetic sequence: The formula used to find any term (a_n) in the sequence without listing all previous ones, given by a_n = a_1 + (n-1)d.
Sum of arithmetic series concepts (to be detailed on Page 2): Understanding how to efficiently calculate the total sum of an arithmetic sequence, particularly useful for sequences with many terms.
Quick reference formulas (introduced here):
Fibonacci rule: F_n = F_{n-1} + F_{n-2} with initial terms (e.g., F_1=1, F_2=1 or F_0=0, F_1=1).
General arithmetic sequence components: first term a_1, common difference d, nth term a_n = a_1 + (n-1)d, sum S_n = \frac{n}{2}[2a_1 + (n-1)d] = \frac{n}{2}(a_1 + a_n).
Sigma notation: \sum_{k=1}^n a_k as a compact way to represent a series, where k is the index of summation, 1 is the lower limit (starting term), n is the upper limit (ending term), and a_k is the general term or expression for each term.
Page 2 - Arithmetic sequences & series
nth term: a_n = a_1 + (n-1)d
Example: If a_1 = 5 (first term) and d = 2 (common difference), then the 3rd term (a_3) is 5 + (3-1)2 = 5 + 4 = 9.
Sum of arithmetic series: S_n = \frac{n}{2}[2a_1 + (n-1)d] = \frac{n}{2}(a_1 + a_n)
This formula allows for quick calculation of the sum of the first n terms, often used when n is large, simplifying the summation process.
Real-life relevance: Recognize usefulness of arithmetic sequences and sums in real-life contexts such as calculating consistent savings growth (e.g., adding a fixed amount each month), predicting population growth at a constant rate, or determining seating capacity in an auditorium with rows that increase by a fixed number of seats. They are also used in scenarios involving uniform increase or decrease over time.
Geometric sequences & series
Geometric sequence: A sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r). If r = 0, the sequence becomes a1, 0, 0, ext{…}; if r = 1, it's a constant sequence (a1, a1, a1, ext{…});
nth term: a_n = a_1 r^{\,n-1}, where a_1 is the first term and r is the common ratio (e.g., 2, 6, 18, 54, \ldots where a_1=2, r=3). This formula directly computes any term without needing previous terms.
Sum of geometric series:
Finite sum: S_n = \frac{a_1\left(1 - r^{n}\right)}{1 - r} (for r \neq 1)
Used for calculating total growth or decay over a specific number of periods or terms, such as compound interest calculations over a few years or the total depreciation of an asset.
Infinite sum (when |r| < 1): S = \frac{a_1}{1 - r}
This formula is applicable only when the absolute value of the common ratio is less than 1 (|r| < 1). This condition ensures that the terms of the series progressively get smaller and approach zero, allowing the sum to converge to a finite value. If |r| \ge 1, the terms would either stay the same, grow indefinitely, or oscillate, and the sum would not converge. This is relevant in contexts like calculating the total distance traveled by a bouncing ball until it stops or the total value of perpetual annuities.
Real-life relevance: Understand growth/decay scenarios like compound interest in finance, exponential population growth, radioactive decay of unstable isotopes, or the spread of a disease where the rate of change is proportional to the current amount.
Harmonic sequences
Definition: A sequence whose reciprocals form an arithmetic sequence. If a_1, a_2, a_3, \ldots is a harmonic sequence, then \frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \ldots is an arithmetic sequence. This unique relationship makes them relevant in fields such as physics (e.g., frequencies of