28 arithmetic–geometric-mean inequality

2

arithmetic–geometric-mean inequality See

MEAN

.

arithmetic mean See

MEAN

.

arithmetic sequence (arithmetic progression) A

SE

-

QUENCE

of numbers in which each term, except the first,

differs from the previous one by a constant amount is

called an arithmetic sequence. The constant difference

between terms is called the common difference. For

example, the sequence 4, 7, 10, 13, … is arithmetic with

common difference 3. An arithmetic sequence can be

thought of as “linear,” with the common difference

being the

SLOPE

of the linear relationship.

An arithmetic sequence with first term aand com-

mon difference dhas the form:

a, a + d, a + 2d, a + 3d, …

The nth term anof the sequence is given by an=

a+(n– 1)d. (Thus, the 104th term of the arithmetic

sequence 4,7,10,13,…, for example, is a104 = 4 +

[103 ×3] = 313.)

The sum of the terms of an arithmetic sequence is

called an arithmetic series:

a+ (a+ d) + (a+ 2d) + (a+ 3d)+…

The value of such a sum is always infinite unless the

arithmetic sequence under consideration is the constant

zero sequence: 0,0,0,0,…

The sum of a finite arithmetic sequence can be

readily computed by writing the sum both forward and

backward and summing column-wise. Consider, for

example, the sum 4 + 7 + 10 + 13 + 16 + 19 + 22 + 25

+ 28 + 31. Call the answer to this problem S. Then:

4 + 7 + 10 + 13 + 16 + 19 + 22 + 25 + 28 + 31 = S

31 + 28 + 25 + 22 + 19 + 16 + 13 + 10 + 7 + 4 = S

and adding columns yields:

35 + 35 + 35 + 35 + 35 + 35 + 35 + 35 + 35 + 35 = 2S

That is, 2S= 10 ×35 = 350, and so S= 175. In general,

this method shows that the sum of nequally spaced

numbers in arithmetic progression, a + b + … + y+ z, is

ntimes the average of the first and last terms of the sum:

It is said that C

ARL

F

RIEDRICH

G

AUSS

(1777–1855), as

a young student, astonished his mathematics instructor

by computing the sum of the numbers 1 though 100 in

a matter of seconds using this method. (We have 1 + 2

+ … + 100 = 100 ×= 5,050.)

See also

GEOMETRIC SEQUENCE

;

SERIES

.

arithmetic series See

ARITHMETIC SEQUENCE

.

array An ordered arrangement of numbers or symbols

is called an array. For example, a

VECTOR

is a one-

dimensional array: it is an ordered list of numbers. Each

number in the list is called a component of the vector. A

MATRIX

is a two-dimensional array: it is a collection of

numbers arranged in a finite grid. (The components of

such an array are identified by their row and column

positions.) Two arrays are considered the same only if

they have the same number of rows, the same number of

columns, and all corresponding entries are equal. One

can also define three- and higher-dimensional arrays.

In computer science, an array is called an identifier,

and the location of an entry is given by a subscript. For

example, for a two-dimensional array labeled A, the

entry in the second row, third column is denoted A23.

An n-dimensional array makes use of nsubscripts.

A

_ryabhata (ca. 476–550

C

.

E

.) Indian Trigonometry,

Number theory, Astronomy Born in Kusumapura,

now Patna, India, A

–ryabhata (sometimes referred to as

A

–ryabhata I to distinguish him from the mathematician

of the same name who lived 400 years later) was the

first Indian mathematician of note whose name we

know and whose writings we can study. In the section

Ganita (Calculation) of his astronomical treatise

A

–ryabhatiya, he made fundamental advances in the the-

ory of

TRIGONOMETRY

by developing sophisticated

techniques for finding and tabulating lengths of half-

chords in circles. This is equivalent to tabulating values

of the sine function. A

–ryabhata also calculated the

value of πto four decimal places (π≈62,832/20,000 =

3.1416) and developed rules for extracting square and

1 + 100

2