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

  , where

  a

  is the first term,

  n

  is the term number, and

  d

  is the common difference.
• Geometric sequences can be represented as

  a * r^(n - 1)

  , where

  a

  is the first term,

  r

  is the common ratio, and

  n

  is the term number.
• A series can be seen as the sum of a sequence

  S = a + (a + d) + (a + 2d) + ...

  , where

  S

  is the sum of the series,

  a

  is the first term and

  d

  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

  , where

  a_n

  is the nth term,

  a

  the first term,

  n

  the term number, and

  d

  the common difference.
• To find particular terms in a geometric sequence, you can use the formula

  a_n = a * r^(n - 1)

  , where

  a_n

  is the nth term,

  a

  is the first term,

  r

  is the common ratio, and

  n

  is the term number.
• The sum of an arithmetic series can be found using

  S_n = n/2 * [2a + (n - 1)d]

  , where

  S_n

  is the sum of the first n terms, and

  a

  ,

  d

  , and

  n

  are as defined above.
• The sum of a geometric series for n terms is calculated by

  S_n = a * (1 - r^n) / (1 - r)

  for

  r ≠ 1

  . For an infinite geometric series where

  -1 < r < 1

  , the sum

  S

  is determined by

  S = 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.