4.5.1 Language of Sequences & Series

1. Sequences

A sequence is a list of numbers. This list may or may not have a pattern between terms. If there is a pattern, it can be defined in two ways:

As an nth term.
As an inductive/recursive/recurrence relation.

$N^{th}$ Term

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Example: Find the first 4 terms of $u_n = 7n + n^2$ . Note: $u_1$ means term $1, u_k$ means term $k$ .

u_1 \text{ is when } n=1 : \quad u_1 = 7(1) + (1)^2 = 8

u_2 = 7(2) + (2)^2 = 18

u_3 = 7(3) + (3)^2 = 30

u_4 = 7(4) + (4)^2 = 44

2. Term

A term is an individual element in a sequence. For a sequence $\{a_n\}$ , the $n$ th term is denoted $a_n$ .

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Example:

3. Common Difference (Arithmetic Sequences)

The common difference is the constant difference between consecutive terms in an arithmetic sequence.

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Example:

4. Common Ratio (Geometric Sequences)

The common ratio is the constant factor between consecutive terms in a geometric sequence.

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Example:

5. Series

A series is the sum of the terms of a sequence. If the sequence is infinite, the series is also called an infinite series.

Notation: The sum of the first $n$ terms of a sequence $\{a_n\}$ is denoted by $S_n$ , and it is called a partial sum.

$S_n = a_1 + a_2 + \dots + a_n$

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6. Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence.

Sum of the first $n$ terms:

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

$S_n = \frac{n}{2} \left( a_1 + a_n \right)$

where a is the first term, $d$ is the common difference, and $n$ is the number of terms.

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Example:

7. Geometric Series

A geometric series is the sum of the terms of a geometric sequence.

Sum of the first $n$ terms:

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

where $a$ is the first term and $r$ is the common ratio.

Sum of an infinite geometric series (when $|r| < 1$ ):

$S = \frac{a}{1 - r}$

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Example:

For the sequence $2, 4, 8, 16, \dots$ , the sum of the first 3 terms is:

$S_3 = 2 \times \frac{1 - 2^3}{1 - 2} = 2 \times \frac{1 - 8}{-1} = 2 \times 7 = 14$

8. Convergence and Divergence

Convergent Series: An infinite series is said to converge if the sum approaches a finite value as the number of terms increases.

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Example: The series $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$ converges to $1$ .

Divergent Series: An infinite series diverges if the sum increases without bound or oscillates as more terms are added.

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Example: The series $1 + 2 + 3 + 4 + \dots$ diverges, as it tends to infinity.

9. Sigma Notation (Summation Notation)

Sigma notation is a compact way to write the sum of a sequence. It uses the Greek letter $\Sigma$ (sigma) to represent the sum.

Notation:

$\sum_{i=1}^{n} a_i$

This means "sum the terms $a_i$ from $i = 1$ to $n$ ."

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Example:

$\sum_{i=1}^{4} i$ means $1 + 2 + 3 + 4 = 10$ .

Summary:

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Sequence: An ordered list of numbers following a specific pattern.
Term: An individual number in a sequence.
Series: The sum of the terms of a sequence.
Arithmetic Sequence/Series: A sequence/series with a constant difference between terms.
Geometric Sequence/Series: A sequence/series with a constant ratio between terms.
Convergence/Divergence: Whether an infinite series sums to a finite value or not.
Sigma Notation: A compact way to represent the sum of a sequence.

Language of Sequences & Series Simplified Revision Notes for A-Level AQA Maths Pure

4.5.1 Language of Sequences & Series

1. Sequences

2. Term

3. Common Difference (Arithmetic Sequences)

4. Common Ratio (Geometric Sequences)

5. Series

6. Arithmetic Series

7. Geometric Series

8. Convergence and Divergence

9. Sigma Notation (Summation Notation)

Summary:

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