Wp/lus/chhiarna-khaidiat

Wikifiahbu lamah hêngte hi ...

En bawk la : chhiarna-khaidiat, inpuntir-khaidiat

Chhiarna-khaidiat kan tih chu nambar inzui

$a_1, a_2, a_3, a_3, ...$

a inkar hlatzawng inchen chho thliah thliah hi a ni.

Entirna

Entirnan a hnuaia zâtte khu en ta la:

$3, 5, 7, 9, 11$

3 leh 5 inthlauhna chu 2; 5 leh 7 inthlauhna pawh pahnih; 7 leh 9 inthlauhna pawh 2; 9 leh 11 inthlauhna pawh pahnih a ni a, chuvangin chhiarna-inzui

$3, 5, 7, 9, 11$

hi chhiarna-khaidiat a ni. A zât indawtte inthlauhna, hetah chuan 2, hi inthlauhtlànna kan ti. Amaherawhchu a hnuaia zât inzuite khu en ta la: $1, 5, 8, 9, 77$, a zât indawtte inthlauhna a inan loh avangin chhiarna-khaidiat a ni lo a ni.

Chhiarna-khaidiat lär ber chu

$1, 2, 3, 4, 5, 6, 7, ...$

hi a ni. A dang lar leh deuhte chu

$5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, ...$

tih te,

$10, 20, 30, 40, 50, 60, 70, ...$

tih te hi a ni.

Chhiarna-khaidiat belh dàn

Nambar 1 aṭanga 100 thleng hi belh duh ta la, khawlthluakneia chhut luh chhung pawh a buaithlâk tham a, chuvangin a tih dàn awlsam hriat a ngai a ni. 1 aṭanga 100 hi chhiarna-khaidiat a nih miau avangin belh dàn awlsam tak a awm a, lo tilang ta ila:

$1+2+3+...+100=100\frac{1+100}{2}$

a ni a, a dinglama mi khi chu tu pawhin awlsam takin kan belh thei tawh ang. Chuvangin a huapzovin sawi ta ila, chhiarna-khaidiat 1 aṭanga n thleng

$a_1, a_2, ..., a_n$

lo nei ta ila, a inthlauhtlànna chu d lo ni ta se, avai belhkhâwm chu

$a_1+a_2+...+a_n=\frac{a_1+a_n}{2}$

emaw

$a_1+...+a_n=n\frac{2a_1+(n-1)d}{2}$

a lo ni ta a ni.

Belhna dàn lo chhuah dàn

A chunga belh awlsam dàn khi heti hian awlsam takin kan hmu thei ang. Chhiarna-khaidiatte belh kha B tiin lo vuah ta ila:

$B=a_1+a_2+ ... +a_n$

Chhiarna-khaidiata a zât hmasa ber kha $a_1$ a nih chuan a dawt chu $a_1$-a d (inthlauhtlànna) belh: $a_1+d$ a lo ni a, a dawt leh chu $a_1$-a d vawi hnih belh: $a_1+d+d=a_1+2d$ a lo ni leh a, chutiang zëlin. Kawng lehlamah chuan a tawp ber $a_n$ leh a tawp ber dawttu $a_{n-1}$ inthlauhna chu d a ni a; $a_n$ hi $a_{n-1}$-a d belh a ni; ṭawngkam danga sawiin $a_{n-1}$ chu $a_n$ aṭanga d paih a ni:$a_{n-1}=a_n-2$. A tawp ber dawttu, dawt lehtu $a_{n-2}$ pawh $a_n$ aṭanga d vawi hnih paih: $a_{n-2}=a_n-d-d=a_n-2d$, chutiang zëlin. Chuvangin zât inzuite kha heti hian kan ziak thei a:

$a_1, a_1+d, ... , a_1+(n-1)d$;

chu chu a lêt zawngin heti hian kan ziak thei bawk ang:

$a_n, a_n-d, a_n-2d, ... , a_n-(n-1)d$.

Tichuan B chuan ziah dàn chi hnih a nei ta:

$B=a_1+(a_1+d)+\cdots +(a_1+(n-1)d)$

$B=a_n+(a_n-d)+\cdots+(a_n-(n-1)d)$

Hemite pahnih hi lo belh ta ila, a d leh -d, ... , (n-1)d leh -(n-1)d te a inthai chhe ta vek a, a chuangbäng awm chhun chu $a_1+a_n$ vawi n inbelh khawm a ni. Chumi awmzia chu

$B+B=\underbrace{a_1+a_n+\cdots}_{vawi\ n}$

tihna a ni a, chu chu ziah dan danga ziahin:

$2B=n{a_1+a_n}$

tihna a ni a, 2 hi dinglamah sawn phei ta dâwrh ila intlukna thar

$B=n\frac{a_1+a_n}{2}$

kan lo dawng ta a. Heta ṭang hian $a_n=a_1+(n-1)d$ tih kan hman chuan

$B=n\frac{2a_1+(n-1)d}{2}$

tih kan lo dawng ta a ni. He mite pahnih hi a ni, a chunga kan tihlan kha ni.

Chhiarna-khaidiat belhna chanchin

A chunga kan tihlan tâk belh dàn khi kum AD 499-a a lehkhabu ziah Aryabhatiya-ah Aryabhata, India hunhlui chhiarkawpmi ropui tak chuan a lo ziak lang tawh a ni.