Sequence and Series PDF

vuqØe rFkk Js.kh 143 vè;k; 8 vuqØe rFkk Js.kh (Sequence and Series) v“Natural numbers are the product of human spirit” – Dedekind v 8.1 Hkwfedk (I...

vuqØe rFkk Js.kh 143 vè;k; 8 vuqØe rFkk Js.kh (Sequence and Series) v“Natural numbers are the product of human spirit” – Dedekind v 8.1 Hkwfedk (Introduction) xf.kr esa] 'kCn ^vuqØe* dk mi;ksx lk/kj.k v¡x”sz kh osQ leku fd;k tkrk gSA tc ge dgrs gSa fd lewg osQ vo;oksa dks vuqØe esa lwphc¼ fd;k x;k gS rc gekjk rkRi;Z gS fd lewg dks bl izdkj Øfed fd;k x;k gS fd ge mlosQ lnL;ksa dks izFke] f}rh;] r`rh; la[;k rFkk vkfn ls igpku ldrs gSaA mnkgj.kr%] fofHkUu le;ksa esa ekuo dh tula[;k vFkok cSDVhfj;k vuqØe dh jpuk djrs gSAa dksbZ /ujkf'k tks cSad [kkrsa esa tek dj nh tkrh gS] fofHkUu o"kksZa esa ,d vuqØe dk fuekZ.k djrh gSA fdlh lkeku dh voewfY;r dhersa ,d vuqØe cukrh gSa ekuo fØ;kvksa osQ dbZ {ks=kksa esa vuqØeksa dk cgqr egÙoiw.kZ mi;ksx gSA fof'k"V iSVuks± dk vuqlj.k djus okys Fibonacci vuqØe Js.kh (Progression) dgykrs gSaA fiNyh d{kk esa] ge (1175-1250 A.D.) lekarj Js.kh osQ laca/ esa i 1 osQ fy, an −1 12. a1 = –1, an = , tgk¡ n ≥ 2 n 13. a1 = a2 = 2, an = an–1–1, tgk¡ n > 2 14. Fibonacci vuqØe fuEufyf[kr :i esa ifjHkkf"kr gS % 1 = a1 = a2 rFkk an = an–1 + an–2, n.>2 rks an +1 an Kkr dhft,] tcfd n = 1, 2, 3, 4, 5 8.4 xq.kksÙkj Js.kh [Geometric Progression (G. P.)] vkb, fuEufyf[kr vuqØeksa ij fopkj djsa % (i) 2,4,8,16,.... 1 −1 1 −1 (ii) , , , ,... 9 27 81 243 (iii).01,0001,.000001,... bues ls izR;sd vuqØe osQ in fdl izdkj c