arithmetic

(adjective)

Of a progression, mean, etc, computed using addition rather than multiplication.

Related Terms

Examples of arithmetic 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.
    • An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.
    • 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.
    • The general form for an infinite arithmetic series is:

  • 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

    • 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.
    • There is a trick that can be used to add up the terms of any 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:

  • 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:

  • Basic Operations

    • The basic arithmetic operations for real numbers are addition, subtraction, multiplication, and division.
    • Addition is the most basic operation of arithmetic.

  • Simplifying Algebraic Expressions

    • Algebraic expressions may be simplified, based on the basic properties of arithmetic operations.
    • Algebraic operations work in the same way as arithmetic operations, such as addition, subtraction, multiplication, division and exponentiation and are applied to algebraic variables and terms.
    • Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations (addition, subtraction, multiplication, division and exponentiation).

  • Introduction to Sequences

    • 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 just start writing out the terms.

  • Geometric Sequences

    • 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.

  • Negative Exponents

    • Numbers with negative exponents are treated normally in arithmetic operations and can be rewritten as fractions.