2.4

Sequences

10 flashcards to master Sequences

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Definition Flip

Define a 'sequence' in mathematics and provide an example.

Answer Flip

A sequence is an ordered list of numbers (or other elements) called terms. Each term follows a specific pattern or rule.

Example: 2, 4, 6, 8, ... is a sequence of even numbers.
Key Concept Flip

What is the 'nth term' of a sequence and how is it useful?

Answer Flip

The 'nth term' is a formula that allows you to calculate any term in the sequence directly based on its position (n). It's useful for finding specific terms without listing all the preceding terms.

Example: For sequence 2n+1, the 5th term is 2(5)+1 = 11.
Key Concept Flip

Find the next two terms in the sequence: 3, 7, 11, 15, ...

Answer Flip

This is an arithmetic sequence with a common difference of 4. The next two terms are 19 (15 + 4) and 23 (19 + 4).

Definition Flip

Explain the difference between an arithmetic and a geometric sequence.

Answer Flip

An arithmetic sequence has a constant difference between consecutive terms (addition/subtraction). A geometric sequence has a constant ratio between consecutive terms (multiplication/division).

Definition Flip

What is the 'common difference' in an arithmetic sequence, and how do you find it?

Answer Flip

The 'common difference' is the constant value added to each term to get the next term in an arithmetic sequence. You find it by subtracting any term from the term that follows it.

Example: In the sequence 1, 4, 7, 10..., the common difference is 4 - 1 = 3.
Key Concept Flip

The 4th term of an arithmetic sequence is 14 and the 7th term is 23. Find the common difference.

Answer Flip

The difference between the 7th and 4th term (23 - 14 = 9) spans 3 common differences. Therefore, the common difference is 9 / 3 = 3.

Definition Flip

What is the 'common ratio' in a geometric sequence, and how do you find it?

Answer Flip

The 'common ratio' is the constant value multiplied by each term to get the next term in a geometric sequence. You find it by dividing any term by the term that precedes it.

Example: In the sequence 2, 6, 18, 54..., the common ratio is 6 / 2 = 3.
Key Concept Flip

Determine the nth term of the following arithmetic sequence: 5, 8, 11, 14, ...

Answer Flip

The common difference is 3. The nth term is of the form 3n + c. Substitute n = 1: 3(1) + c = 5, so c = 2. Therefore, the nth term is 3n + 2.

Definition Flip

Explain what a quadratic sequence is and give an example.

Answer Flip

A quadratic sequence is one where the nth term is a quadratic expression (e.g., an^2 + bn + c). The difference between consecutive terms is not constant, but the difference between those differences is constant.

Example: 1, 4, 9, 16, ... (n^2).
Key Concept Flip

The second term of a geometric sequence is 6 and the fourth term is 24. Find the possible values of the common ratio.

Answer Flip

Let the first term be 'a' and the common ratio be 'r'. Then ar = 6 and ar^3 = 24. Dividing the second equation by the first, we get r^2 = 4. Therefore, r = 2 or r = -2.

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2.3 Inequalities 2.5 Quadratics

Key Questions: Sequences

Define a 'sequence' in mathematics and provide an example.

A sequence is an ordered list of numbers (or other elements) called terms. Each term follows a specific pattern or rule.

Example: 2, 4, 6, 8, ... is a sequence of even numbers.
Explain the difference between an arithmetic and a geometric sequence.

An arithmetic sequence has a constant difference between consecutive terms (addition/subtraction). A geometric sequence has a constant ratio between consecutive terms (multiplication/division).

What is the 'common difference' in an arithmetic sequence, and how do you find it?

The 'common difference' is the constant value added to each term to get the next term in an arithmetic sequence. You find it by subtracting any term from the term that follows it.

Example: In the sequence 1, 4, 7, 10..., the common difference is 4 - 1 = 3.
What is the 'common ratio' in a geometric sequence, and how do you find it?

The 'common ratio' is the constant value multiplied by each term to get the next term in a geometric sequence. You find it by dividing any term by the term that precedes it.

Example: In the sequence 2, 6, 18, 54..., the common ratio is 6 / 2 = 3.
Explain what a quadratic sequence is and give an example.

A quadratic sequence is one where the nth term is a quadratic expression (e.g., an^2 + bn + c). The difference between consecutive terms is not constant, but the difference between those differences is constant.

Example: 1, 4, 9, 16, ... (n^2).

About Sequences (2.4)

These 10 flashcards cover everything you need to know about Sequences for your Cambridge IGCSE Mathematics (0580) exam. Each card is designed based on the official syllabus requirements.

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