Math Analysis & Approaches

Numbers & Algebra

# Sequences & Series

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Sequences & Series

**Sequences & Series**

*Definition of Sequences and Series*

- A
**sequence**is an ordered list of numbers following a specific pattern. An explicit formula is often used to generate the terms in a sequence. - A
**series**is the sum of the terms of a sequence. It is represented as the sum of the sequence terms with a specified start and end term.

*Understanding Sequences and Series*

- Sequences can be
**finite**, with a specific number of terms, or**infinite**, with an endless number of terms. Finite sequences often appear when calculating the terms of a pattern or list, while infinite sequences frequently occur in computational mathematics to approximate numbers. - There are two primary types of sequences in the Numbers & Algebra topic:
**arithmetic sequences**and**geometric sequences**. An arithmetic sequence adds (or subtracts) the same number each time to form the next term, while a geometric sequence multiplies (or divides) by the same number. - Arithmetic sequences can be represented with the formula
a + (n - 1) * d

, wherea

is the first term,n

is the term number, andd

is the common difference. - Geometric sequences can be represented as
a * r^(n - 1)

, wherea

is the first term,r

is the common ratio, andn

is the term number. - A
**series**can be seen as the sum of a sequenceS = a + (a + d) + (a + 2d) + ...

, whereS

is the sum of the series,a

is the first term andd

the common difference.

*Finding Terms in a Sequence and Sum of a Series*

- To find particular terms in an arithmetic sequence, you can use the formula
a_n = a + (n - 1) * d

, wherea_n

is the nth term,a

the first term,n

the term number, andd

the common difference. - To find particular terms in a geometric sequence, you can use the formula
a_n = a * r^(n - 1)

, wherea_n

is the nth term,a

is the first term,r

is the common ratio, andn

is the term number. - The sum of an arithmetic series can be found using
S_n = n/2 * [2a + (n - 1)d]

, whereS_n

is the sum of the first n terms, anda

,d

, andn

are as defined above. - The sum of a geometric series for n terms is calculated by
S_n = a * (1 - r^n) / (1 - r)

forr ≠ 1

. For an infinite geometric series where-1 < r < 1

, the sumS

is determined byS = a / (1 - r)

.

*Use of Sequences and Series*

- Sequences and series are heavily used in many aspects of mathematics and practical applications. These could include calculating total distance in alternating movement scenarios, calculating total motorway toll fees over a period, or determining accumulated interest in financial contexts.
- Understanding these core concepts also lays the foundation for more complex mathematics involving calculus and infinite series.

*Convergent and Divergent Series*

- A
**convergent series**is an infinite series whose sum is a finite number, while a**divergent series**is an infinite series that does not have a finite sum. This can often be seen in geometric series, where a geometric series converges if-1 < r < 1

. - Recognising whether a series is convergent or divergent can be essential for solving mathematical problems and applying these concepts in practical situations. For instance, evaluating whether a sum of money paid over time will exceed a certain amount.