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**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.

**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.

**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.

**Understanding Basic Terminology**:

Define and distinguish between a sequence and a series.

Identify terms of a sequence using U_{n} 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).

**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.

- 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.

March 12, 2024

Planes of Symmetry in 3D Shapes for Key Stage 3/GCSE students.

Use isometric paper for hands-on learning and enhanced understanding.

March 8, 2024

Master GCSE Math: Get key SOH-CAH-TOA tips, solve triangles accurately, and tackle area tasks. Ideal for students targeting grades 4-5.

March 7, 2024

Explore Regions in the Complex Plane with A-Level Further Maths: inequalities, Argand diagrams, and geometric interpretations.