arithmetic

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.

• Mean: The Average

  • When we think of means, or averages, we are typically thinking of the arithmetic mean.
  • For example, per capita income is the arithmetic average income of a nation's population.
  • The arithmetic mean is defined via the expression:
  • Comparison of the arithmetic, geometric and harmonic means of a pair of numbers.
  • Define the average and distinguish between arithmetic, geometric, and harmonic means.

• Log Transformations

  • This occurs because, as shown below, the anti-log of the arithmetic mean of log-transformed values is the geometric mean.Table 1 shows the logs (base 10) of the numbers 1, 10, and 100.
  • The arithmetic mean of the three logs is
  • Therefore, if the arithmetic means of two sets of log-transformed data are equal then the geometric means are equal.

• 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:

• The Sample Average

  • The sample average (also called the sample mean) is often referred to as the arithmetic mean of a sample, or simply, $\bar{x}$ (pronounced "x bar").
  • For a finite population, the population mean of a property is equal to the arithmetic mean of the given property while considering every member of the population.
  • The arithmetic mean is the "standard" average, often simply called the "mean".
  • For example, the arithmetic mean of five values: 4, 36, 45, 50, 75 is:
  • The mean is the arithmetic average of a set of values, or distribution; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely (mode).

• 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).