Arithmetic progression - Tousasuuretsu

Japanese: 等差数列 - とうさすうれつ

A sequence that is created by adding a certain number to another number one by one. It is also called an arithmetic progression, and is sometimes written as AP (arithmetic progression). The first number is the first term, and the constant numbers added one by one are called the common difference. If the first term is a and the common difference is d, then the nth term, a _n , is:
a _n =a+(n-1)d
In particular, when three numbers _a1 , _a2 , and _a3 form an arithmetic progression, the number _a2 in between is called the mean arithmetic term. The mean arithmetic term is expressed as the arithmetic mean of the two numbers _a1 and _a3 at both ends. That is,

If the sum of an arithmetic progression with first term a, common difference d, and number of terms n is S _n , then

Here, l represents the last term of the sequence (called the terminal term). In particular, the sum of the nth term of a sequence of odd numbers 1, 3, 5, … is ⁿ² . And for a sequence _b1 , _b2 , … whose general term is a quadratic function of the term number n, that is, _bn = ^An2 + Bn + C, the difference sequence is an arithmetic sequence, and if C = 0, then _bn is the sequence that represents the sum of the first n terms of an arithmetic sequence _a1 , _a2 , …. The reciprocal of the sequence _a1 , _a2 , …, that is,

When a ₁ , a ₂ , … form an arithmetic sequence, a 1, a ₂ , … are called a harmonic sequence. In particular, when three numbers a ₁ , a 2, a ₃ form a harmonic sequence, a ₂ is called the mean harmonic. This is the harmonic mean of a ₁ and a _3.

It is.

[Osamu Takenouchi]

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

ある数に、一定の数を次々に加えていってできる数列。算術数列ともいい、A. P.（arithmetic progression）と書くこともある。最初の数を初項、次々に加える一定数を公差という。初項をa、公差をdとするとき、その第n項a_nは、
　　a_n=a+(n-1)d
と表される。とくに、三つの数a₁,a₂,a₃が等差数列をなすとき、間の数a₂を等差中項という。等差中項は両端の数a₁、a₃の相加平均（算術平均）で表される。すなわち

である。初項a、公差d、項数nの等差数列の和をS_nとすれば、

である。ここでlは、この数列の最後の項（末項という）を表す。とくに、奇数からなる数列1,3,5,……のn項の和はn²である。そして、一般項が項の番号nについての二次式であるような数列b₁,b₂,……すなわちb_n=An²+Bn+Cであるような数列については、その階差数列は等差数列であり、もしここでC=0ならば、b_nは、ある等差数列a₁,a₂,……に対して、そのn項目までの和を表す数列となる。数列a₁,a₂,……の逆数、すなわち

が等差数列をなすとき、a₁,a₂,……を調和数列という。とくに、三つの数a₁,a₂,a₃が調和数列をなすとき、a₂を調和中項という。これはa₁、a₃の調和平均

である。