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ljleGg ul0ftLo wf/0ff÷ljifoj:t'aLr ;DaGw :yflkt x'g] tyf ;d:ofsf] cy{k"0f{ tl/sfn]
;dfwfg ug{ ;lsg] x'gfn] o; ljlwnfO{ ul0ft lzIf0fdf lgs} dxŒjk"0f{ dflgG5 .
lgudg ljlw
of] cfudg ljlwsf] ljkl/t ljlw xf] . o;df ;fwf/0faf6 ljlzi6lt/, efjfTdsaf6 7f];lt/,
;"qaf6 pbfx/0flt/ ts{sf] k|jfx x'g] ub{5 . ;a} ul0ftsf] clGtd ¿k g} lgudg ¿k xf] .
lzIfsn] sIffdf lzIf0f ug] { kf7l;t ;DalGwt ;"q / ;f]sf] k|of]u s;/L ug] { eGg]af/]df
hfgsf/L lbP/ ;f]xLcg';f/ ;d:of ;dfwfg ÷lx;fa ug{ nufpg] ljlw g} lgudg ljlw xf] .
cfudgfTds ts{ k|0ffnLsf] ;'? pbfx/0fx¿af6 x'G5 / w]/} pbfx/0fx¿df kfOg] ;dfg
u'0fx¿sf] cfwf/df Pp6f lgisif{ lgsflnG5 . of] lgisif{ g} lgudgfTds ts{ k|0ffnLsf nflu
ul0ftLo l;4fGt aG5 / o;}sf] k|of]u u/L ul0ftLo ;d:ofx¿sf] xn ul/G5 .
;a} ul0ftLo wf/0ffx¿sf] lzIf0f ubf{ cfudg / lgudg ljlwsf] k|of]u ;fy;fy} ug'{kb{5
. klxn]
cfudg ljlwaf6 wf/0ff :ki6 kfl/;s] kl5 lgudg ljlwaf6 ;f] wf/0ffsf] ;anLs/0fsf nflu
cEof; u/fpg' kb{5 . ul0ft lzIf0fdf klg oxL s'/fnfO{ cfTd;ft ug'{
kb{5 .
cfudg ljlwaf6 lahf]/ ;ª\Vofx¿sf] of]ukmnsf] ju{d"n lgsfNg] Pp6f pbfx/0f x]/f} ― .
oxf―, s]xL lahf]/ ;ªVofx¿sf] of]ukmnsf] ju{d"n lgsfNb} hfcf} ― M
1 = 1
1+3 = 4 = 2
1+3+5 = 9 = 3
1+3+5+7 = 16 = 4
1+3+5+7+9 = 25 = 5
................................................
1+3+5+7+9+....+99 = ?
ca 1 b]lv 99 ;Ddsf lahf]/ ;ª\Vofx¿sf] of]ukmnsf] ju{d"n lgsfNgsf nflu 99 ;Ddsf ;a}
lahf]/ ;ªVof hf]8]/ ju{d"n lgsfNg ;do a9L nfUg] x'G5 . t;y{, ;'¿sf s]xL ;ª\Vofsf]
ju{d"n cfpg] ] 9f―rfsf] cjnf]sg ug{ nufO{ 5nkmn u/L 9f―rf kQf nufpg] . oxf―, hltcf]6f
ljhf]/ ;ª\Vof -kb_ sf] of]ukmnsf] ju{d"n lgsfNg] xf] ToxL ;ª\Vof -kbsf] ;ª\Vof_ g} ju{d"n
cfpg] /x]5 eGg] ;fdfGoLs/0f ug{ nufpg] .
ca, 1 b]lv 99 ;Ddf hDdf 50 cf]6f lahf]/ ;ª\Vofx¿ x'G5g\ . t;y{,
1+3+5+7+9+....+99 = 50 x'G5 eGg] lgisif{ lgsfNg nufpg] .
lgudg ljlwsf] pbfx/0f M
(a 2 - b 2 ) = (a +b) (a -b) x'G5 eGg] lgisif{af6 c¿ aLhLo cleJo~hs, dfgf} ― 16x 2 - 25y 2 sf],
v08Ls/0f ug{ ;lsG5 . o;sf nflu klxn] 16x 2 - 25y 2 nfO{ (4x) 2 - (5y) 2 agfO{ (a 2 - b 2 ) sf]
:j¿kdf nuL o;nfO{ (a 2 - b 2 ) = (a +b) (a -b) ;―u t'ngf ubf{, (4x) 2 - (5y) 2 = (4x +5y) (4x -5y)
x'G5 .
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