Sequences and Series

Prerequisite Knowledge

  1. Linear Sequences:
    • Understanding the concept of a sequence.
    • Finding the nth term of a linear sequence.
    • Generating terms of a sequence given the nth term.
  2. Geometric Progressions (GPs):
    • Recognizing and distinguishing a geometric progression.
    • Finding the common ratio of a GP.
    • Generating terms of a GP given the first term and the common ratio.
    • Continuing a given GP for a specified number of terms.
  3. Arithmetic Progressions (APs):
    • Recognizing and distinguishing an arithmetic progression.
    • Finding the common difference of an AP.
    • Generating terms of an AP given the first term and the common difference.

Success Criteria

Understanding Basic Terminology:

Define and distinguish between a sequence and a series.

Identify terms of a sequence using Un​ notation.

Recognize arithmetic and geometric sequences/series.

Arithmetic Sequences:

Determine the common difference of an arithmetic sequence.

Use the formula Un = a + (n – 1)d to find the nth term.

Sum terms using the formula

S_n=\frac{n}{2}(2a\ +\ (n\ -\ 1)\ d)

Geometric Sequences:

Determine the common ratio of a geometric sequence.

Use the formula

U_n=a*r^{(n-1)}

to find the nth term.

Sum terms using:

S_n=\frac{a(1-r^n)}{1-r}

Convergence:

Determine whether a sequence or series is convergent or divergent. Find the sum to infinity of a convergent geometric series using

S_\infty=\frac{a}{1-r}

Recurrence Relations:

Understand and derive the terms of a sequence from a given recurrence relation.

Solve problems using initial terms and recurrence relations.

Sigma Notation (Σ):

Recognize and expand sequences and series expressed using sigma notation.

Apply arithmetic operations and properties of sigma notation.

Application to Real-world Scenarios:

Model and solve real-world problems using sequences and series (e.g., compound interest, population growth).

Key Concepts

Foundational Knowledge:

Understand the difference between sequences (an ordered list of numbers) and series (the sum of the terms of a sequence).

Sequences:

  • An ordered list of numbers is called a sequence.
  • The terms of a sequence can be finite or infinite.
  • Common sequences include arithmetic sequences, geometric sequences, and Fibonacci sequences.

Series:

  • A series is the sum of the terms of a sequence.
  • Infinite series can be convergent or divergent.
  • Common series include arithmetic series, geometric series, and telescoping series.

Recurrence sequences:

  • In a recurrence sequence, each term is defined in terms of the previous terms.
  • Recurrence sequences can be used to model various phenomena, such as population growth, radioactive decay, and the stock market.

Convergence and divergence:

  • A series is convergent if the sum of its terms approaches a finite number as the number of terms tends to infinity.
  • A series is divergent if the sum of its terms does not approach a finite number, as the number of terms tends to infinity.

Common Misconceptions

  • Students often confuse sequences and series. A sequence is an ordered list of numbers, while a series is the sum of the terms of a sequence.
  • Students often forget that the terms of a sequence can be finite or infinite. Infinite sequences are often more difficult to work with than finite sequences.
  • Students often make mistakes when calculating the sum of a series. It is important to be careful when adding up large numbers, and to use correct rounding procedures.
  • Students often forget that infinite series can be convergent or divergent. A convergent series has a finite sum, while a divergent series does not.
  • Students often make mistakes when using recurrence sequences. Recurrence sequences can be difficult to understand, and it is important to be careful when solving for the terms of a recurrence sequence.

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