Arifmeti chna aritmeti chna 1 progre siya ce poslidovnist dijsnih chisel kozhen chlen yakoyi pochinayuchi z drugogo utvoryuyetsya dodavannyam do poperednogo chlena odnogo j togo zh chisla Zagalnij vid arifmetichnoyi progresiyi a 1 a 1 d a 1 2 d a 1 n 1 d displaystyle a 1 a 1 d a 1 2d ldots a 1 n 1 d ldots de a 1 displaystyle a 1 ce pershij chlen progresiyi d a n 1 a n displaystyle d a n 1 a n Chislo d displaystyle d nazivayut rizniceyu arifmetichnoyi progresiyi Arifmetichna progresiya ye monotonnoyu poslidovnistyu Yaksho d gt 0 displaystyle d gt 0 to vona zrostaye a pri d lt 0 displaystyle d lt 0 vona spadaye Yaksho d 0 displaystyle d 0 to progresiya ye staloyu Zmist 1 Znahodzhennya UNIQ postMath 00000008 QINU go chlena arifmetichnoyi progresiyi 2 Vlastivist arifmetichnoyi progresiyi 3 Suma UNIQ postMath 00000023 QINU pershih chleniv arifmetichnoyi progresiyi 3 1 Suma UNIQ postMath 00000024 QINU poslidovnih chleniv pochinayuchi z pershogo chlena 3 2 Suma UNIQ postMath 0000002C QINU poslidovnih chleniv pochinayuchi z k go chlena 3 3 Suma pershih UNIQ postMath 00000035 QINU naturalnih chisel 4 Div takozh 5 Primitki 6 Posilannya na storonni dzherela 7 DzherelaZnahodzhennya n displaystyle n go chlena arifmetichnoyi progresiyi RedaguvatiDlya usih chleniv progresiyi pochinayuchi z drugogo spravedliva rivnist a n a n 1 d displaystyle a n a n 1 d nbsp Za oznachennyam arifmetichnoyi progresiyi a 2 a 1 d displaystyle a 2 a 1 d nbsp a 3 a 2 d a 1 d d a 1 2 d displaystyle a 3 a 2 d a 1 d d a 1 2d nbsp a 4 a 3 d a 1 2 d d a 1 3 d displaystyle a 4 a 3 d a 1 2d d a 1 3d nbsp a 5 a 4 d a 1 3 d d a 1 4 d displaystyle a 5 a 4 d a 1 3d d a 1 4d ldots nbsp Prostezhuyetsya zakonomirnist a n a 1 n 1 d displaystyle a n a 1 n 1 d nbsp Dovedennya Dovedemo pravilnist gipotezi dlya vsih n N displaystyle n in mathbb N nbsp za dopomogoyu metodu matematichnoyi indukciyi Dlya n 1 displaystyle n 1 nbsp a 1 a 1 1 1 d a 1 displaystyle a 1 a 1 1 1 d a 1 nbsp Pripustimo sho n k displaystyle n k nbsp a k a 1 k 1 d displaystyle a k a 1 k 1 d nbsp Dovedemo sho formula pravilna dlya n k 1 displaystyle n k 1 nbsp tobto sho pravilne nastupne a k 1 a 1 k d displaystyle a k 1 a 1 kd nbsp Dlya cogo vikoristayemo pripushennya a k 1 a k d a 1 k 1 d d a 1 k d d d a 1 k d displaystyle a k 1 a k d a 1 k 1 d d a 1 kd d d a 1 kd nbsp Otzhe formula n displaystyle n nbsp go chlena maye viglyad n N displaystyle forall n in mathbb N nbsp a n a 1 n 1 d displaystyle a n a 1 n 1 d nbsp Vlastivist arifmetichnoyi progresiyi RedaguvatiVirazimo chleni a n 1 displaystyle a n 1 nbsp ta a n 1 displaystyle a n 1 nbsp cherez a n displaystyle a n nbsp i d displaystyle d nbsp a n 1 a n d displaystyle a n 1 a n d nbsp i a n 1 a n d displaystyle a n 1 a n d nbsp Znajdemo yihnye serednye arifmetichne a n 1 a n 1 2 a n d a n d 2 a n a n 2 a n displaystyle frac a n 1 a n 1 2 frac a n d a n d 2 frac a n a n 2 a n nbsp Tobto bud yakij chlen arifmetichnoyi progresiyi pochinayuchi z drugogo ye serednim arifmetichnim dvoh susidnih chleniv n 2 displaystyle forall n geq 2 nbsp a n a n 1 a n 1 2 displaystyle a n frac a n 1 a n 1 2 nbsp Suma n displaystyle n pershih chleniv arifmetichnoyi progresiyi RedaguvatiSuma n displaystyle n nbsp poslidovnih chleniv pochinayuchi z pershogo chlena Redaguvati Zapishemo sumu poslidovnih chleniv arifmetichnoyi progresiyi dvoma sposobami S n a 1 a 1 d a 1 n 2 d a 1 n 1 d displaystyle S n a 1 a 1 d ldots a 1 n 2 d a 1 n 1 d nbsp S n a 1 n 1 d a 1 n 2 d a 1 d a 1 displaystyle S n a 1 n 1 d a 1 n 2 d ldots a 1 d a 1 nbsp Dodamo ci dva virazi 2 S n 2 a 1 n 1 d 2 a 1 n 1 d 2 a 1 n 1 d 2 a 1 n 1 d displaystyle 2S n 2a 1 n 1 d 2a 1 n 1 d ldots 2a 1 n 1 d 2a 1 n 1 d nbsp 2 S n n 2 a 1 n 1 d displaystyle 2S n n 2a 1 n 1 d nbsp Podilimo obidvi chastini na 2 S n 2 a 1 n 1 d 2 n a 1 a 1 n 1 d 2 n a 1 a n 2 n displaystyle S n frac 2a 1 n 1 d 2 n frac a 1 a 1 n 1 d 2 n frac a 1 a n 2 n nbsp Otzhe suma n displaystyle n nbsp pershih chleniv arifmetichnoyi progresiyi mozhe buti virazhena takimi formulami S n i 1 n a i a 1 a n 2 n 2 a 1 d n 1 2 n displaystyle S n sum i 1 n a i a 1 a n over 2 n 2a 1 d n 1 over 2 n nbsp Suma n displaystyle n nbsp poslidovnih chleniv pochinayuchi z k go chlena Redaguvati Iz arifmetichnoyi progresiyi a 1 a 2 a 3 a k a k 1 a k 2 a k n 1 displaystyle a 1 a 2 a 3 ldots a k a k 1 a k 2 ldots a k n 1 ldots nbsp mozhna vidiliti pidposlidovnist b n a k n 1 displaystyle b n a k n 1 nbsp sho ye arifmetichnoyu progresiyeyu Todi suma n displaystyle n nbsp pershih chleniv b n displaystyle b n nbsp S n b 1 b n 2 n a k a k n 1 2 n displaystyle S n b 1 b n over 2 n a k a k n 1 over 2 n nbsp Otzhe suma n displaystyle n nbsp poslidovnih chleniv arifmetichnoyi progresiyi pochinayuchi z k displaystyle k nbsp go chlena S n a k a k n 1 2 n displaystyle S n a k a k n 1 over 2 n nbsp Suma pershih n displaystyle n nbsp naturalnih chisel Redaguvati nbsp Animovane dovedennya formuli dlya znahodzhennya sumi pershih n naturalnih chiselSumu pershih n displaystyle n nbsp naturalnih chisel mozhna zapisati yak 1 2 n S n 2 1 n 1 1 2 n 1 n 2 n n n 1 2 displaystyle 1 2 cdots n S n 2 cdot 1 n 1 cdot 1 over 2 n 1 n over 2 n n n 1 over 2 nbsp Otzhe suma pershih n displaystyle n nbsp naturalnih chisel 1 2 n n n 1 2 displaystyle 1 2 cdots n frac n n 1 2 nbsp Cya formula vidoma yak trikutne chislo Isnuye istoriya 2 pro te yak Karl Gauss vidkriv cyu formulu koli navchavsya u tretomu klasi Shob podovshe zajnyati ditej uchitel poprosiv klas porahuvati sumu pershih sta chisel 1 2 99 100 displaystyle 1 2 dots 99 100 nbsp Gauss pomitiv sho poparni sumi z protilezhnih kinciv odnakovi 1 100 101 displaystyle 1 100 101 nbsp 2 99 101 displaystyle 2 99 101 nbsp tosho i tomu zmig vidrazu vidpovisti sho suma dorivnyuye 5050 displaystyle 5050 nbsp Dijsno legko bachiti sho rishennya zvoditsya do formuli n n 1 2 displaystyle frac n n 1 2 nbsp tobto do formuli sumi pershih n displaystyle n nbsp chisel naturalnogo ryadu Div takozh RedaguvatiMonotonna poslidovnist Matematichna indukciya Generalna sukupnist Geometrichna progresiya Multisekciya ryadu Uzagalnena arifmetichna progresiya Figurni chisla Kombinatorika Rekurentne spivvidnoshennya Gipoteza Erdesha pro arifmetichni progresiyiPrimitki Redaguvati 123 Bukvospoluchennya th u slovah greckogo pohodzhennya Ukrayinskij pravopis ukrayinskoyu Ukrayinska nacionalna komisiya z pitan pravopisu 2019 Arhiv Ukrayinskij pravopis originalu za 17 veresnya 2019 Procitovano 29 sichnya 2021 Gauss s Day of Reckoning American Scientist angl 6 lyutogo 2017 Procitovano 23 zhovtnya 2022 Posilannya na storonni dzherela RedaguvatiArifmetichni poslidovnosti na Mathworld Arhivovano 4 chervnya 2011 u Wayback Machine angl FIZMA neT matematika onlajn Arhivovano 15 travnya 2021 u Wayback Machine Dzherela RedaguvatiKorn G Korn T Spravochnik po matematike dlya nauchnih rabotnikov i inzhenerov Moskva Nauka 1970 720 s 100000 prim ros Tropfke J Arithmetik und Algebra Berlin Walter de Gruyter 1980 755 s nim nbsp Ce nezavershena stattya z matematiki Vi mozhete dopomogti proyektu vipravivshi abo dopisavshi yiyi Otrimano z https uk wikipedia org w index php title Arifmetichna progresiya amp oldid 40310840
