sequence

Examples of sequence in the following topics:

• Sequences

  • A sequence is an ordered list of objects (or events).
  • Also, the sequence $(1, 1, 2, 3, 5, 8)$, which contains the number $1$ at two different positions, is a valid sequence.
  • Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers $(2, 4, 6, \cdots)$.
  • Finite sequences are sometimes known as strings or words, and infinite sequences as streams.
  • The empty sequence $( \quad )$ is included in most notions of sequence, but may be excluded depending on the context.

• Arithmetic Sequences

  • An arithmetic sequence is a sequence of numbers in which the difference between the consecutive terms is constant.
  • An arithmetic progression, or arithmetic sequence, is a sequence of numbers such that the difference between the consecutive terms is constant.
  • For instance, the sequence $5, 7, 9, 11, 13, \cdots$ is an arithmetic sequence with common difference of $2$.
  • The behavior of the arithmetic sequence depends on the common difference $d$.
  • Calculate the nth term of an arithmetic sequence and describe the properties of arithmetic sequences

• DNA Sequencing Techniques

  • The Sanger sequencing method was used for the human genome sequencing project, which was finished its sequencing phase in 2003, but today both it and the Gilbert method have been largely replaced by better methods.
  • When the human genome was first sequenced using Sanger sequencing, it took several years, hundreds of labs working together, and a cost of around $100 million to sequence it to almost completion.
  • Sanger sequence can only produce several hundred nucleotides of sequence per reaction.
  • Most next-generation sequencing techniques generate even smaller blocks of sequence.
  • Most genomic sequencing projects today make use of an approach called whole genome shotgun sequencing.

• DNA Sequencing Based on Sanger Dideoxynucleotides

  • Sanger sequencing, also known as chain-termination sequencing, refers to a method of DNA sequencing developed by Frederick Sanger in 1977.
  • More recently, dye-terminator sequencing has been developed.
  • Automated DNA-sequencing instruments (DNA sequencers) can sequence up to 384 DNA samples in a single batch (run) in up to 24 runs a day.
  • Automation has lead to the sequencing of entire genomes.
  • Different types of Sanger sequencing, all of which depend on the sequence being stopped by a terminating dideoxynucleotide (black bars).

• Strategies Used in Sequencing Projects

  • The strategies used for sequencing genomes include the Sanger method, shotgun sequencing, pairwise end, and next-generation sequencing.
  • All of the segments are then sequenced using the chain-sequencing method.
  • A larger sequence that is assembled from overlapping shorter sequences is called a contig.
  • This is the principle behind reconstructing entire DNA sequences using shotgun sequencing.
  • Compare the different strategies used for whole-genome sequencing:  Sanger method, shotgun sequencing, pairwise-end sequencing, and next-generation sequencing

• Sequences of Mathematical Statements

  • In mathematics, a sequence is an ordered list of objects, or elements.
  • Unlike a set, order matters in sequences, and exactly the same elements can appear multiple times at different positions in the sequence.
  • A sequence is a discrete function.
  • Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers $(2,4,6, \cdots )$.
  • Sequences of statements are necessary for mathematical induction.

• Introduction to Sequences

  • Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers $(2, 4, 6, \cdots )$.
  • Finite sequences are sometimes known as strings or words and infinite sequences as streams.
  • Finite sequences include the empty sequence $( \quad )$ that has no elements.
  • These are called recursive sequences.
  • Assume our sequence is $t_1, t_2, \dots $.

• The General Term of a Sequence

  • Given terms in a sequence, it is often possible to find a formula for the general term of the sequence, if the formula is a polynomial.
  • Given several terms in a sequence, it is sometimes possible to find a formula for the general term of the sequence.
  • Then the sequence looks like:
  • Then the sequence would look like:
  • The second sequence of differences is:

• DNA Sequencing of Insertion Sites

  • An insertion sequence (also known as an IS, an insertion sequence element, or an IS element) is a short DNA sequence that acts as a simple transposable element.
  • The coding region in an insertion sequence is usually flanked by inverted repeats.
  • Although insertion sequences are usually discussed in the context of prokaryotic genomes, certain eukaryotic DNA sequences belonging to the family of Tc1/mariner transposable elements may be considered to be insertion sequences.
  • A complex transposon does not rely on flanking insertion sequences for resolvase.
  • It involves cutting genomic DNAs with a restriction enzyme, ligating vectorettes to the ends, and amplifying the flanking sequences of a known sequence using primers derived from the known sequence along with a vectorette primer.

• Geometric Sequences

  • The $n$th term of a geometric sequence with initial value $a$ and common ratio $r$ is given by
  • Generally, to check whether a given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio.
  • The common ratio of a geometric series may be negative, resulting in an alternating sequence.
  • For instance: $1,-3,9,-27,81,-243, \cdots$ is a geometric sequence with common ratio $-3$.
  • The behavior of a geometric sequence depends on the value of the common ratio.