arithmetic sequence

Examples of arithmetic sequence in the following topics:

• 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

• Summing Terms in an Arithmetic Sequence

  • An arithmetic sequence which is finite has a specific formula for its sum.
  • The sum of the members of a finite arithmetic sequence is called an arithmetic series.
  • An infinite arithmetic series is exactly what it sounds like: an infinite series whose terms are in an arithmetic sequence.
  • Even if one is dealing with an infinite sequence, the sum of that sequence can still be found up to any $n$th term with the same equation used in a finite arithmetic sequence.
  • Calculate the sum of an arithmetic sequence up to a certain number of terms

• Introduction to Sequences

  • 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.
  • An arithmetic (or linear) sequence is a sequence of numbers in which each new term is calculated by adding a constant value to the previous term.
  • An explicit definition of an arithmetic sequence is one in which the $n$th term is defined without making reference to the previous term.
  • To find the explicit definition of an arithmetic sequence, you begin writing out the terms.

• Recursive Definitions

  • When discussing arithmetic sequences, you may have noticed that the difference between two consecutive terms in the sequence could be written in a general way:
  • In this equation, one can directly calculate the nth-term of the arithmetic sequence without knowing the previous terms.
  • An applied example of a geometric sequence involves the spread of the flu virus.
  • Using this equation, the recursive equation for this geometric sequence is:
  • Use a recursive formula to find specific terms of a sequence

• Applications and Problem-Solving

  • Arithmetic series can simplify otherwise complex addition problems by decreasing the number of terms to be added.
  • Using equations for arithmetic sequence summation can greatly facilitate the speed of problem solving.
  • This trick applies to all arithmetic series.
  • As long as you go up by the same amount as you go down, the sum will stay the same—and this is just what happens for arithmetic series.

• Sums and Series

  • Summation is the operation of adding a sequence of numbers; the result is their sum or total.
  • If you add up all the terms of an arithmetic sequence (a sequence in which every entry is the previous entry plus a constant), you have an arithmetic series.
  • You understand that this trick will work for any arithmetic series.
  • If we apply this trick to the generic arithmetic series, we get a formula that can be used to sum up any arithmetic series.
  • Every arithmetic series can be written as follows:

• Geometric Sequences

  • Generally, to check whether a given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio.
  • 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.
  • Geometric sequences (with common ratio not equal to $-1$, $1$ or $0$) show exponential growth or exponential decay, as opposed to the linear growth (or decline) of an arithmetic progression such as $4, 15, 26, 37, 48, \cdots$ (with common difference $11$).
  • Note that the two kinds of progression are related: exponentiating each term of an arithmetic progression yields a geometric progression, while taking the logarithm of each term in a geometric progression with a positive common ratio yields an arithmetic progression.

• 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.

• 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.

• Converting Units

  • For example, 10 miles per hour can be converted to meters per second by using a sequence of conversion factors.
  • So, when the units mile and hour are cancelled out and the arithmetic is done, 10 miles per hour converts to 4.47 meters per second.