Sequence - Suuretsu

Japanese: 数列 - すうれつ

A sequence of numbers a 1 , a 2 , a 3 , a 4 , … arranged in correspondence with each of the natural numbers ₁ , ₂ , ₃ , ₄ , … is called a sequence. Each of these numbers is called a term of the sequence, and from the beginning, they are called the first term, the second term, the third term, …. When the nth term of a sequence is considered to be representative of each term, it is called the general term. The general term is often given as an expression for n . There are finite sequences, which end with a finite number of terms, and infinite sequences, which continue indefinitely. The last term of a finite sequence is called the terminal term. Arithmetic sequences and geometric sequences are often used as examples of sequences. However, there are many other types of sequences as well.

Given a sequence of numbers a ₁ , a ₂ , …, the sum of the first to nth terms is often a problem. This is called

To define a sequence of numbers a ₁ , a ₂ , …, each a _n may be defined as an expression of the previous terms a ₁ , a ₂ , …, a _{n -1} .

a _n = f ( a ₁ , a ₂ ,……, a _{n -1} )
In this case, this formula is called a recurrence formula. The recurrence formula for an arithmetic progression is a _n = a _{n -1} + d ( d is crossover).
The recurrence formula for a geometric progression is a _n = a _{n -1} r ( r is the common ratio).
The Fibonacci sequence is expressed by the following recurrence formula.

a ₁ = 1, a ₂ = 1, a _n = a _{n -1} + a _{n -2}
( n ≧3)
For a sequence of numbers a ₁ , a ₂ ,……,
a _n ⁽¹⁾ = a _{n +1} - a _n
( n = 1, 2, ...)
The sequence a ₁ ⁽¹⁾ , a ₂ ⁽¹⁾ , … defined by is called a first-order difference sequence,
a _n ⁽²⁾ = a _{n +1} ⁽¹⁾ - a _n ⁽¹⁾
( n = 1, 2, ...)
The sequence a ₁ ⁽²⁾ , a ₂ ⁽²⁾ , … determined by is called a second-difference sequence, and similarly it is called a third-difference sequence, … It can be useful to create and study a difference sequence in order to understand the structure of a sequence.

The limit of an infinite sequence is a discussion of what happens as we go further into it. Sometimes we investigate what happens when we add to an infinite sequence starting from the first term. A sequence a ₁ ＋ a ₂ ＋ ……, which is formally connected by _a plus sign to form _a series , is called a series.

[Osamu Takenouchi]

Sums of commonly used sequences
©Shogakukan ">

Sums of commonly used sequences

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

自然数1, 2, 3, 4,……のおのおのに対応して並べられた数の列a₁, a₂, a₃, a₄,……を数列という。そのおのおのの数を数列の項といい、初めから順に、初項、第2項、第3項、……という。数列の第n項をおのおのの項の代表と考えるとき、これを一般項という。一般項は、nの式として与えられることが多い。数列には、項が有限個で終わる有限数列と、どこまでも続く無限数列とがある。有限数列の最後の項を末項という。数列の例としては等差数列、等比数列などがよく扱われる。しかし、そのほかにも、いろいろな数列がある。

　数列a₁, a₂,……が与えられたとき初項から第n項までの和がよく問題になる。これを記号

で表す。数列a₁, a₂,……を定めるのに、各a_nをそれより以前の項a₁, a₂,……, a_n-1の式として定めることがある。

　　a_n＝f(a₁, a₂,……, a_n-1)
このとき、この式を漸化式という。等差数列の漸化式は
　　a_n＝a_n-1＋d　（dは交差）
等比数列の漸化式は
　　a_n＝a_n-1r　（rは公比）
次の漸化式で表されるのがフィボナッチ数列である。

　　a₁＝1,　a₂＝1,　a_n＝a_n-1＋a_n-2
　　(n≧3)
　数列a₁, a₂,……に対して、
　　a_n⁽¹⁾＝a_n+1－a_n
　　(n＝1, 2,……)
によって定められる数列a₁⁽¹⁾, a₂⁽¹⁾,……を第一階差数列、
　　a_n⁽²⁾＝a_n+1⁽¹⁾－a_n⁽¹⁾
　　(n＝1, 2,……)
によって定められる数列a₁⁽²⁾, a₂⁽²⁾,……を第二階差数列といい、同様に第三階差数列、……という。数列の成り立ちを知るのに階差数列をつくって調べると役にたつことがある。