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 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.

• 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:

• Averages

  • The arithmetic mean, or average, of a set of numbers indicates the "middle" or "typical" value of a data set.
  • The arithmetic mean, or "average" is a measure of the "middle" or "typical" value of a data set.
  • The arithmetic mean is used frequently not only in mathematics and statistics but also in fields such as economics, sociology, and history.
  • For example, per capita income is the arithmetic mean income of a nation's population.
  • The arithmetic mean $A$ is defined via the expression: