Веленкин;Комбинаторика
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ÌÒÆcÒÔ |
3( ) = ² vin iyrk€ Ív {véuysn krsí·ntqp q qxrsí·ntqp wvsy·jnu − − + = lvwyxzqu€n wnénxzjtvkrq
bs¹ xsvkj ©rvxuvxª xzvs•rv n bs¹ xsvkj ©zjézjéª qunnu− + − = lvwyxzqu€} wnénxzjtvkvr
Æ ¡nxzqotj·tvn ·qxsv uvmyz kvpzq vltj lkn qsq zéq wjé€ vlqtjrvk€} |q{é Ìlty wjéy uv tv k€iéjz• & xwvxvijuq
ïqxsv wnénxzjtvkvr qo éjosq·t€} q vlqtjrvk€} |q{é éjktv 3( ) = = Êo tq} k = wnénxzjtvkrj} lkn vlq
tjrvk€n |q{é€ qlyz wvlé¹l dtj·qz k ëzvu xsy·jn wvsy·jnu( − ) = ¡nxzqotj·t€} ·qxns bkn wjé€ vlqtjrvk€} |q{é
uv tv k€iéjz• & |
= xwvxvijuq wvxsn ·nmv & = xwvxvijuq uv tv |
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k€iéjz• n•n lkn |q{é€ Ìi•nn ·qxsv wnénxzjtvkvr ëzq} |q{é éjktv
3( ) = wéq·nu k = qo tq} nxz• }vz¹ i€ vltj wjéj
xsnlyí•q} léym oj léymvu vlqtjrvk€} |q{é j k = wnénxzjtvkrj}
² lkn zjrqn wjé€ Ív {véuysn krsí·ntqq q qxrsí·ntqp wvsy·jnu ·zv ëzvz xsy·jp ljnz ( − + ) = ty t€} tju ·qxns
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¬tjsvmq·tv tj}vlqu ·zv zéq |
wjé€ |
vlqtjrvk€} |
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= ty t€} tju ·qxns Æxnmv ·qxns |
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Ìi•nn rvsq·nxzkv w¹zqotj·t€} ·qxns rvzvé€n uv |
tv xvxzj |
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kqz• qo ljtt€} |q{é éjktv + & & |
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Êo q} ·qxsj k 3 |
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= |q{éj qlnz wvlé¹l zéq éjoj Ívsy |
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·jnu qxrvu€} ·qxns
Ìi•nn ·qxsv wnénxzjtvkvr ljtt€} |q{é éjktv 3( ) Êo tq} k 3( ) wnénxzjtvkrj} ljttj¹ |q{éj xzvqz lkj éjoj wvlé¹l k 3( ) qlyz wjéjuq ljtt€n |q{é€ k 3( ) ² ljtt€n|q{é€ q k 3( ) ² ljtt€n |q{é€ Ív {véuysn krsí·ntqp q qxrsí·ntqp wvsy·jnu ·zv tqrjrqn |q{é€ tn wvkzvé¹ízx¹ k
3( ) − 3( ) + 3( ) − − 3( ) + 3( ) =
wnénxzjtvkrj} bs¹ ·qxsj jtjsvmq·tv wvsy·jnu ·zv ·qxsv |
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wnénxzjtvkvr éjktv |
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− + = i |
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− q yknsq·qkjnu xénltqp |
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&P− k€iqéjnu zéq tvunéj qo P |
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tj ivs•¡qp ² tj |
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Q − & Q + & Q − |
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cxsq isvr kxzén·jnzx¹ zv nmv uv tv ojuntqz• vltvp tvkvp iyrkvp Înzvlvu krsí·ntqp q qxrsí·ntqp wvsy·jnu vzknz
-sjkj ,,
3( ) − 3( ) − 3( ) − 3( ) + |
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+ 3( ) + 3( ) + 3( ) − 3( ) |
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U+ = Q + |
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U U |
= Q &U− |
zv xsnkj wvsy·jnzx¹ |
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rjr &Q+ |
U + &Q |
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U U + |
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) xwéjkj |
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Q− |
U(U + ) |
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(Q + ) |
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− Q |
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U(U + ) |
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j ïqxsv éjoun•ntqp x wvkzvéntq¹uq qo Q ësnuntzvk wv
éjktv Q èjovi•nu ëzq éjoun•ntq¹ tj rsjxx€ vztnx¹ r N uy rsjxxy éjoun•ntq¹ xvlné j•qn évktv N éjosq·t€} zqwvk ësnuntzvk ïqxsv
éjoun•ntqp wnékvmv rsjxxj éjktv & kzvévmv ² & wvxsn k€ivéj |
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lky} zqwvk ësnuntzvk qunnzx¹ xwvxvij k€ivéj ësnuntzj k}vl¹•nmv
lkj éjoj k éjoun•ntqn q wvxsn ëzvmv zéq xwvxvij k€ivéj unxzj ls¹ vlqtv·tvmv ësnuntzj j ·qxsv éjoun•ntqp zénz•nmv rsjxxj éjktv
&Q3 Ìzxílj k€znrjnz wnékvn xvvztv¡ntqn
i ¬tjsvmq·t€u viéjovu éjoiqkjnu tj rsjxx€ éjoun•ntq¹ x wv kzvéntq¹uq xvlné j•qn }vz¹ i€ vlqt ësnuntz {qrxqévkjttvmv zqwj
΀ wvsy·jnu ·zv (Q + ) − Q = + & + & vzrylj x y·nzvu j q |
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xsnlynz lvrjo€kjnuvn xvvztv¡ntqn |
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k m bvrjo€kjízx¹ zv·tv zjr n tv éjxxujzéqkjízx¹ éjoun•ntq¹ x wvkzvéntq¹uq qo ësnuntzvk Q zqwvk wv ësnuntzj k rj lvu
èjxxuvzéqu kxn xv·nzjtq¹ x wvkzvéntq¹uq qo ësnuntzvk Q zq
Q
wvk wv Q Ê} ·qxsv éjktv & Q− èjovi•nu ëzq xv·nzjtq¹ tj rsjxx€
vztnx¹ r N uy rsjxxy xv·nzjtq¹ k rvzvé€n k}vl¹z ësnuntz€ évktv N Q−N xv·nzjtqp u€ k€iqéjnu
N éjosq·t€} zqwvk Æ N p rsjxx k}vl¹z &Q&Q−
&N xwvxvijuq N zqwvk ësnuntzvk k}vl¹•q} k xv·nzjtq¹ ëzvmv rsjxxj |
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Q−N Q xv·nzjtqp x |
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j qo ësnuntzvk ljtt€} N zqwvk uv |
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tv xvxzjkqz• &Q− |
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wvkzvéntq¹uq k rvzvé€n |
k}vl¹z ësnuntz€ kxn} N |
zqwvk |
Ívëzvuy |
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= & &Q− + & &Q− + + &Q& |
Æ€éj j¹ ·qxsj |
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·néno {jrzvéqjs€ wéq}vlqu r lvrjo€kjnuvuy xvvztv¡ntqí
bvrjo€kjnuvn éjkntxzkv uv tv ojwqxjz• zjr
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− & &U− |
+ & &U− |
− = &U |
&Q+U− |
Q Q+U− |
Q Q+U− |
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bs¹ lvrjojzns•xzkj kvo•unu kxn U xv·nzjtq¹ x wvkzvéntq¹uq qo ësnuntzvk Q zqwvk q tjplnu lkyu¹ xwvxvijuq ·qxsv kxn} zjrq} xv·n zjtqp xvxzv¹•q} sq¡• qo ësnuntzvk éjosq·t€} zqwvk f vltvp xzv
évt€ ëzv ·qxsv éjktv &U f léymvp xzvévt€ ·qxsv U xv·nzjtqp |
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wvkzvéntq¹uq qo ësnuntzvk Q zqwvk k rvzvé€n wv réjptnp unén lkj |
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U− N |
Òjr rjr ëzq |
N |
l€ k}vl¹z ësnuntz€ ljtt€} N zqwvk éjktv &Q+U− N− |
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ÌÒÆcÒÔ |
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zqwvk uv tv k€iéjz• &NQ xwvxvijuq zv wéqunt¹¹ {véuysy krsí·ntqp
q qxrsí·ntqp wéq}vlqu r lvrjo€kjnuvuy xvvztv¡ntqí
èjovi•nu kxn wnénxzjtvkrq qo P ins€} q Q ·nét€} ¡jévk tj rsjxx€ Ç rsjxxy (N N NP) vztnxnu wnénxzjtvkrq k rvzvé€} nxz•
N qovsqévkjttv xzv¹•qp ins€p ¡jé N wjé N zévnr NP é¹lvu
xzv¹•q} P ins€} ¡jévk ‰xtv ·zv N + N + + PNP = P fvx·qzjnu ·qxsv wnénxzjtvkvr rsjxxj (N N NP) cxsq Q ·né
t€} ¡jévk wvxzjksnt€ k é¹l zv unxz rylj uv tv xzjkqz• ins€n ¡jé€ iylnz Q + Êo ëzq} unxz N unxz vrj yzx¹ ojt¹z€ vltqu
ins€u ¡jévu N ² lkyu¹ NP unxz ² P ins€uq ¡jéjuq q xkv ivlt€uq vxzjtyzx¹ Q − N − − NP + unxz Ívëzvuy ·qxsv xwvxvivk
éjxwénlnsqz• unxzj ls¹ ins€} ¡jévk z n ·qxsv wnénxzjtvkvr rsjxxj (N N NP) éjktv 3(N N NP Q − N − − NP + ) Ívxrvs•ry
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P |
vi•nn ·qxsv wnénxzjtvkvr P ins€} q Q ·nét€} ¡jévk éjktv &Q+P |
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wvsy·jnu lvrjo€kjnuvn xvvztv¡ntqn |
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j Ívsv |
qu 6 |
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= & + & + & + + Q&Q Æ |
xqsy éjkntxzkj |
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&Q−N qunnu 6 |
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&Q− frsjl€kj¹ |
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&N = |
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= Q& + (Q − )& + (Q − )& + |
+ |
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wvsy·jnu 6 |
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= Q(& + & + & + + &Q) q wvzvuy 6 |
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Q− Q |
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n viéjovu yxzjtjksqkjnu ·zv 6 |
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k 6 |
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= Q− (Q − ) + m 6 |
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n 6 |
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= (& + & + & + + Q&Q) − (& + |
& + & + + &Q) = |
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= Q+ Q − Q + |
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΀ qunnu &N = &N− + &N |
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+ & ) − (& |
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) + + (− )QQ&Q− = |
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êzj xyuuj éjktj wéq Q = q wéq Q > |
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q Òjr rjr &N = |
(N + )(N + ) |
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N+ zv ëzj xyuuj x y·nzvu k éjktj |
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(Q + )(Q + ) &Q+ |
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tqn k rkjléjzt€} xrvirj} éjktv
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-sjkj ,, |
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s 6 |
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= nxsq Q tn·nztv q 6 |
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= (− )N&N nxsq Q = N ·nztv bs¹ |
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lvrjojzns•xzkj tjlv wnénutv qz• éjosv |
ntq¹ ( + [)Q q ( − [)Q wvxsn |
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·nmv tjpzq rvë{{q|qntz wéq [Q
Çj lj¹ xzévrj zénymvs•tqrj ¹ks¹nzx¹ jéq{unzq·nxrvp wév ménxxqnp wéq·nu xyuuj éjktvyljsntt€} vz rvt|vk ësnuntzvk xzévrq lnsqzx¹ tj lvrjojzns•xzkv wévkvlqzx¹ x wvuv••í qtlyr|qq wv tvunéy xzévrq dtj·qz ëzqu xkvpxzkvu visjljnz q xyuuj lky} ësn untzvk wénlwvxsnltnp xzévrq z n wvxsnltqp ësnuntz zjisq|€
ftj·jsj k€inénu rjrvp qo zén} wjxxj qévk rvzvé€u inoéjo sq·tv rjr xqlnz• x¹lnz sq|vu r svrvuvzqky êzvz k€ivé uv tv xln
sjz• & = xwvxvijuq Ëj rj lvp xrju•n uv tv wnénxj qkjz• wjxxj |
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qévk 3 = xwvxvijuq Æxnmv wvsy·jnu ( ) = xwvxvivk |
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ftj·jsj k€iqéjnu $ |
xwvxvijuq jivtntzvk |
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tntzvk uv |
tv wnénxzjks¹z• ktyzéq rj |
lvp wjé€ j zjr |
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wjé wvëzvuy vi•nn |
·qxsv |
xwvxvivk tjlv |
éjolnsqz• tj |
= Æxnmv wvsy·jnu |
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j $ = i & = k cxsq lkn lnky¡rq ojknlvuv |
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iylyz wéqmsj¡nt€ tj zjtn| zv qunnzx¹ $ kjéqjtzvk k€ivéj q} wjéz tnévk vxzjk¡qnx¹ ítv¡np k€iqéjíz wjéztné¡ qo ·qxsj lnky¡nr
·zv uv nz i€z• xlnsjtv $ |
xwvxvijuq j kxnmv qunnu $ |
$ = |
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xwvxvivk Ëjrvtn| nxsq ljtt€n lkn lnky¡rq wéqmsj¡nt€ tj zjtn| zv n•n w¹z• lnky¡nr uv tv k€iéjz• & xwvxvijuq
ïnz€én} lnky¡nr uv tv k€iéjz• & xwvxvijuq Ívxsn ëzvmv k€iqéjnu ítv¡np $ xwvxvijuq olnx• y n xy•nxzknt wvé¹lvr Æxnmv
& $ = xwvxvivk
Înxzj ls¹ tn·nzt€} |q{é uv tv k€iéjz• & = xwvxvijuq
Ëj rj lvu unxzn uv nz xzv¹z• vltj qo |q{é sqiv ·nztj¹ sqiv
tn·nztj¹ Æxnmv wvsy·jnu & = ·qxns Êo tq} ·jxz•
tj·qtjnzx¹ x tys¹ Ìxzjnzx¹ ·qxns
cxsq k€iévxqz• tysq zv vxzjtnzx¹ vltv qo ·qxns
Ìxzjsvx• éjxwvsv qz• ëzq |q{é€ tj unxzj} zjr ·zvi€ tj wnékvu unxzn xzv¹s tn tys• bs¹ ëzv uv tv xlnsjz• vltqu xwvxvivu
ls¹ q ² wv & = xwvxvivk kzvéj¹ |q{éj xzjkqzx¹ tj vltv qo
unxz ls¹ ² & = xwvxvijuq Æxnmv + + = ·qxns |
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bs¹ kxn} ·qxns vz lv vzknz wvsy·jnzx¹ k kqln |
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Çvsq·nxzkv ·qxns ojwqx€kjnu€} ino qxwvs•ovkjtq¹ |q{é€ éjktv = Ívëzvuy |q{éy xvlné jz − = ·qxsv
èvktv éjoj |q{éj k}vlqz k ·qxns tjlv sq¡• k€iéjz• vlty qo léymq} |q{é tj vltvu qo unxz
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Ëys• k}vlqz k vltvotj·tvn ___lkyotj·t€} D q zén}
otj·tvn ·qxsv wv ·qxsy kqlj D E q DE ·qxns kqlj D j kxnmv
k ·qxsv bkj l€ tys• k}vlqz k ·qxns kqlj D
Ìin |q{é€ q k}vl¹z k ·qxns nxsq____zénz•¹ |q{éj D vzsq·tj
vz q zv wv kjéqjtzvk kqlvk D D D D j nxsq vtj éjktj qsq zv n•n kjéqjtzj
gq{é€ q k}vl¹z k ·qxsj
j Ìi•nn ·qxsv wjé rvzvé€n uv tv xvxzjkqz• qo ·nsvknr
éjktv & = Æ rj lyí zévpry ^D E F` k}vl¹z wjé€ ^D E` ^D F`
q ^E F` Ívëzvuy k zn·ntqn ltnp kxn wjé€ iylyz wénlxzjksnt€ wv vltvuy éjoy Ívxrvs•ry oj ltnp wévplnz ·nsvknr zv rj l€p léym wvxnzqz nmv éjoj z n iylnz y·jxzkvkjz• k zén} zévprj} Æ€
inénu xtj·jsj |
zévprq k |
rvzvé€n |
k}vlqz wnék€p léym |
êzv uv |
tv |
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xwvxvivk éjoiqz• |
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wjé€ Çvmlj ëzq zévprq k€iéjt€ vxzjízx¹ lkn kvouv tvxzq k€ivéj zévnr k rvzvé€n k}vlqz kzvévp mvxz• ² tjwéquné nxsq wnék€p k}vlqz k zévprq ^ ` ^ ` ^ ` zv kzvévp k}vlqz sqiv k zévprq ^ ` ^ ` sqiv k zévprq ^ ` ^ ` Ívxsn ëzvmv éjx wénlnsntqn vxzjs•t€} mvxznp vwénlns¹nzx¹ vltvotj·tv
Ï·qz€kj¹ kvouv tvxz• wnénxzjtvkvr zévnr mvxznp wvsy·jnu= xwvxvivk
( )
i Êo ·nsvknr uv tv xvxzjkqz• & = zévnr qo ·nsvknr ²
& = qo w¹zq ² & = j qo ·nz€én} sq¡• & = zévprq Ìi•nn |
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·qxsv kjéqjtzvk ©éjxwqxjtq¹ª éjktv |
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léym vrj |
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Íéqunt¹¹ {véuysy krsí·ntqp q qxrsí·ntqp wvsy·jnu vzknz k kqln
$ − & $ + & $ |
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k cxsq vlqt qo léyonp wéq}vlqz rj l€p lnt• lkj ino wvkzvévk |
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tn wvsy·qzx¹ zv qo vxzjs•t€} uv |
tv xvxzjkqz• & = wjé Ívëzvuy |
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·qxsv ©éjxwqxjtqpª wéq rvzvé€} vlqt q zvz |
n ·nsvknr y·jxzkynz |
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n nltnktv éjktv & $ ÌxzDHzF¹ $ |
− & $ xwvxvivk wéqmsj¡ntq¹ |
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Ívxzjkqu |
k xvvzknzxzkqn |
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·qxsv rvxznp x xyuuvp v·rvk Q − U éjktv ·qxsy_rvxznp x xyuuvp v·rvk
Q + U Ìi•nn ·qxsv kxn} rvxznp lvuqtv éjktv & |
= & |
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Çj l€p qo ·nsvknr uv nz qsq kvpzq qsq tn kvpzq k méywwy
kjéqjtzvk Æ€iéjz• N ·nsvknr qo uv tv &N xwvxvijuq cxsq
méywwj tn uv nz i€z• wyxzvp zv wvsy·jnu − & = xwvxvivk
djwqx• DEF votj·jnz ·qxsv ojwqxjttvn |q{éjuq D E F
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uv |
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·qxsv k€iéjtt€} k}vl¹z |
ljtt€n |
lkj |
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cxsq uy wéqmsjxqz k mvxzq |
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·qt éjktv − N Òvmlj |
ntj wéqmsjxqz − N |
nt•qt q |
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xlnsjz• ∑ |
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Ëj snkvu ivézy uvmyz xqlnz• qsq ·nsvknrj qo ·qxsj zn} rvuy inoéjosq·nt k€ivé ivézj cxsq qo q} ·qxsj k€iéjtv N ·nsvknr zv tjlv k€iéjz• n•n − N ·nsvknr qo ·qxsj wénlwv
·qzjí•q} snk€p ivéz Ívxsn ëzvmv vxzjnzx¹ + ( − N) rjtlqljzvk
qo |
rvzvé€} |
k€iqéjnu méni|vk tj wéjk€p ivéz Æxnmv qunnu |
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& −N xwvxvivk k€ivéj fyuuqéy¹ wv N vz N |
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vzknz
ïqxsv uv tv éjoiqz• tj zéq éjosq·t€} xsjmjnu€} zénu¹ xwvxvijuq + + + + + + fyuuj unt•¡j¹ ·nu iylnz
k xsy·j¹} + + = + + = + + = + + = Òjr
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nzvtj uv tv k€tyz• & |
xwvxvijuq zv k & − = xsy·j¹} |
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xyuuj tn unt•¡n |
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ftj·jsj k€inénu unxzj tj rvzvé€} xzvqz |q{éj êzv uv tv |
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|q{é€ |
qsq ·zv uv |
tv xlnsjz• |
xwvxvijuq Æxnmv wvsy·jnu |
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fyuuj |
|q{é |
síivmv qo tjwqxjtt€} ·qxns |
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ly + = q + = Ívëzvuy nxsq ·qxsv ln |
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sn qz un |
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sqzx¹ tj zv xyuuj nmv |q{é éjktj fsnlvkjzns•tv nlqtq|€ q lkvprq ljíz xyuuy êzj xyuuj wvsy·jnzx¹ nxsq ko¹z• nlqtq|€ q lkvprq Êzjr tj¡n ·qxsv xvlné qz nlqtq|€ lkvprq q
zévprq Êo ëzq} |q{é uv |
tv xvxzjkqz• 3( ) = |
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éjosq·t€} ·qxns |
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$ $ ($ − ) + k lky} xsy·j¹} ² rvmlj tj wnék€n |
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unxzj y¡sq kxn xqtqn qsq kxn réjxt€n ¡jé€ tj n unxzv vxzjnzx¹ k€ivé sq¡• qo |knzvk ls¹ Q = uv tv éjxxuvzénz• kxn kjéqjtz€
rvsq·nxzk wv yi€kjtqí vltv|knzt€} ¡jévk vzknz wvsy·qzx¹ k kqln |
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$ 3( ) + $ 3( ) + $ 3( ) + $ 3( ) |
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rjéz qly•q} wvlé¹l vwénlns¹ízx¹ k€ivévu wnékvp ² ëzvkjéqjtzvk qo Îjxz• vwénlns¹nzx¹ k€ivévu qo kvouv t€}
j xwvxvivk i & k
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j i cxz• kjéqjtz€ -f-f- $ $ = xwvxvivk q f-f-f |
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$ $ = |
xwvxvivk vzknz |
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k -f-f- $ $ = xwvxvivk |
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f-f-f $ $ |
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Íyxz• xviéjsqx• ·sntvk vémrvuqznzj Ív yxsvkqí ojlj·q y tq} tnz rsí·j wv réjptnp unén r vltvuy ojury tv vt nxz• y rj lvmv qo vxzjs•t€} ¡nxzq ·sntvk Ívxrvs•ry ëzv wévqx}vlqz wéq síivp
rvuiqtj|qq ·sntvk zv vi•nn ·qxsv ojurvk éjktv & = vi•nn
·qxsv rsí·np éjktv = j rj l€p ·snt vémrvuqznzj qunnz
= rsí·j z n &
j Èyrkj ©vª uv nz k}vlqz• k ·qxsv k€iéjtt€}
qsq éjoj xwvxvivk iyrkj ©rª ² xwvxvijuq q z l Æxnmv wv sy·jnu = rvuiqtj|qp qsq nxsq ©tnxrvs•rvª
² ëzv qsq ivs•¡n tv tn
i ïqxsv rvuiqtj|qp k rvzvé€} kxn zéq iyrk€ éjosq·t€ éjktv
& = rvuiqtj|qp xvlné j•q} évktv éjosq·t€n iyrk€ iylnz
$ = q rvuiqtj|qp xvlné j•q} zvs•rv vlty iyrky nxz• sq¡• |
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Æxnmv xwvxvij k€ivéj |
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k cxsq y·qz€kjz• q wvé¹lvr iyrk zv $ + $ + = xwvxvivk |
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Íévknlntv & = wé¹u€} ïnéno rj lyí zv·ry wév}vl¹z
wé¹u€n fsnlvkjzns•tv qo ëzvp zv·rq k€}vl¹z wnéwntlqrys¹évk èjxxuvzéqu rjrqn sqiv lkn zv·rq tjwéquné $ q % Ínéwntlqrys¹é€ vwy•ntt€n qo zv·rq % tj wé¹u€n wév}vl¹•qn ·néno zv·ry $ wnén xnrjíz kxn wnéwntlqrys¹é€ vwy•ntt€n qo zv·rq $ Êo zv·rq $ k€ }vl¹z wé¹u€n tn wév}vl¹•qn ·néno % dtj·qz qo % tj tq} uv tv vwyxzqz• wnéwntlqrys¹éj Ìtq wnénxnrjízx¹ x wnéwntlqrys¹éjuq vwy•ntt€uq qo zv·rq $ k = zv·rj} Çj l€p qo wnéwntlqry
s¹évk vwy•ntt€} qo zv·rq % tj léymqn wé¹u€n wév}vl¹•qn ·néno $ wnénxnrjnz sq¡• wnéwntlqrys¹évk vwy•ntt€} qo zv·rq $ zjr rjr vt wjéjssnsnt vltvuy qo ëzq} wnéwntlqrys¹évk qiv vwy•nt tj vlty wé¹uyí x tqu Ívsy·jnzx¹ n•n zv·nr
fsnlvkjzns•tv wnéwntlqrys¹é€ vwy•ntt€n qo lky} zv·nr wnén xnrjízx¹ k + = zv·rj} Ëv qo zv·nr uv tv xvxzjkqz•
wjé êzv ljsv i€ = zv·nr wnénxn·ntq¹ tv tnrvzvé€n qo
ëzq} zv·nr xvkwjljíz Êunttv síi€n qo ljtt€} zv·nr viéjoyíz zénymvs•tqr Æ€xvz€ ëzvmv zénymvs•tqrj ¹ks¹í•qnx¹ tnrvzvé€uq qo tj¡q} wnéwntlqrys¹évk wnénxnrjízx¹ k vltvp zv·rn j u€ ëzy zv·ry
y·sq éjoj Òjr rjr zjrq} zénymvs•tqrvk & = zv tjlv vziévxqz• |
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zv·nr q vxzjnzx¹ kvouv t€} zv·nr wnénxn·ntq¹ |
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Æ€inénu wv vltvp sv¡jlq qo rj |
lvp wjé€ $$′ |
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vxzjs•t€} |
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xwvxvivk q k€inénu wvé¹lvr ojwé¹mjtq¹ sv¡jlnp xwvxvivk Æxnmv
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xzv¹z• é¹lvu zv wvé¹lvr iyrk zjrvp f-f-f-f dlnx• u€ qunnu sq¡•wnénxzjtvkvr q & & xsvk
cxsq ·qxsv qoviéj nttvn wnék€uq zénu¹ |q{éjuq éjktv [ zv ·qxsv qoviéj nttvn wvxsnltquq zénu¹ |q{éjuq uv nz wéqtqujz• otj·ntq¹ − [ j kxnmv − [ otj·ntqp Òjr rjr [
unt¹nzx¹ vz lv zv tju tjlv tjpzq xyuuy tjzyéjs•t€} ·qxns vz lv Ìtj éjktj
bs¹ xsvkj ©rv{nkjérjª wvé¹lvr iyrk f-f-f-f-f xvmsjxt€n uv tv wnénxzjks¹z• 3( ) xwvxvijuq j msjxt€n ² 3( )
xwvxvijuq Æxnmv 3( ) 3( ) = xwvxvivk
bs¹ xsvkj ©xjuvkjéª qunnu 3 3( ) = wnénxzjtvkrq
cxz• rvuiqtj|qq xvlné j•qn kxn iyrk€ ©zª ©jª ©éª q rvuiqtj|qq xvlné j•qn wv éjosq·t€n iyrk€ Æxnmv rvuiq tj|qp èjosq·t€} ·nz€én}otj·t€} ·qxns qo |q{é ·qxsj uv
tv xvxzjkqz• 3( ) + 3( ) =
Ív {véuysn krsí·ntqp q qxrsí·ntqp wvsy·jnu ·zv kxn |q
{é€ xvlné jz − & |
+ & |
− & |
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zvs•rv qo |q{é
Èns€n ¡j¡rq uv tv éjxxzjkqz• & xwvxvijuq Ívxsn k€ivéj
wvsnp ls¹ ins€} ¡j¡nr vxzjnzx¹ wvsnp ls¹ ·nét€} tj rvzvé€n q} uv tv wvxzjkqz• & xwvxvijuq Æxnmv & & xwvxvivk béymj¹ svmqrj én¡ntq¹ ljnz vzknz 3( )
& & |
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Ëjlv k€iéjz• qo unxz unxzj ls¹ iyrk€ ©jª êzv uv tv
xlnsjz• & = xwvxvijuq cxsq lvijkqz• yxsvkqn ·zv tqrjrqn lkn
iyrk€ ©jª tn qlyz wvlé¹l zv ls¹ tq} nxz• sq¡• unxzj k ojwqxq
i r t q u€ qunnu & = xwvxvij
cxsq k€iéjtv vltv ·qxsv zv |
kzvévn uv tv k€iéjz• |
qo |
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léymvp ·nztvxzq Ï·qz€kj¹ kvouv |
tvxz• wnénxzjtvkrq ëzq} lky} ·q |
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Éqiv kxn zéq k€iéjtt€} ·qxsj ·nzt€ sqiv vltv ·nztv q lkj |
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tn·nzt€ Ívëzvuy wvsy·jnu & + & & |
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ÌÒÆcÒÔ
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ïnz€én}otj·tvn ·qxsv uv nz xvxzv¹z• sqiv qo ·nz€én} éjo sq·t€} |q{é sqiv qo lky} vlqtjrvk€} q lky} éjosq·t€} |q{é sqiv qo lky} wjé vlqtjrvk€} |q{é Ívëzvuy vi•nn rvsq·nxzkv zjrq} ·qxns
éjktv 3 |
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3( ) + 3( ) + 3( ) + 3( ) =
y·ntqrvk uvmyz yxzjtvkqz• v·nénl• r rj lvuy ërojuntj zvéy xwvxvijuq j r lkyu ërojuntjzvéju ² ( ) xwvxvijuq Íéq
ëzvu k & xsy·j¹} }vz¹ i€ vlqt y·ntqr lvs nt iylnz vltvkén
unttv vzkn·jz• vivqu ërojuntjzvéju k & xsy·j¹} ² lkvn q z l
Ív {véuysn krsí·ntqp q qxrsí·ntqp ·qxsv éjoyut€} xwvxvivk éjx
wénlnsntq¹ éjktv ( ) |
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¬tjsvmq·tv wvsy·jnu ( ) |
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($ ) − & $ ($ ) + & $ ($ ) − & $ ($ ) + & $ ($ ) − & $ ($ ) + & $ |
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Ínékyí mvéqovtzjs• uv tv éjxréjxqz• xwvxvijuq Çj lyí xsnlyí•yí mvéqovtzjs• tjlv éjxréj¡qkjz• zjr ·zvi€ vréjxrj rj lvp rsnzrq vzsq·jsjx• vz vréjxrq sn j•np wvl tnp rsnzrq êzv uv
tv xlnsjz• |
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= xwvxvijuq Ív wéjkqsy |
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wévqoknlntq¹ wvsy·jnu ( ) |
xwvxvivk éjxréjxrq |
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Ëj wvxsnltnu unxzn uv nz i€z• vltj qo |q{é cxsq ojljtj vltj qo ëzq} |q{é zv kzvévn q zénz•n unxzj uv nz ojtqujz•
síij¹ qo ¡nxzq |q{é j wnékvn ² síij¹ qo w¹zq |q{é |
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Æxnmv wvsy·jnu kvouv |
tvxznp dtj·qz xyuuj wv éjoé¹ly nlqtq| |
éjktj ( + ) = |
Òv·tv zjr n tj}vlqu ·zv xyuuj wv |
éjoé¹ly lnx¹zrvk éjktj ( + + + + ) = xvznt ²
z€x¹· ² ( + + + + ) = Æxnmv wv
sy·jnu xyuuy
j Ínék€n zéq |q{é€ ·qxsj wévqokvs•t€ j wvxsnlt¹¹ wéq tqujnz vltv qo lky} otj·ntqp vwénlns¹nu€} vxzjzrvu vz lnsntq¹
xyuu€ wnék€} zén} |q{é tj wvëzvuy ·qxns iylnz
cxsq tj rjrvu zv unxzn ojljz• |q{éy zv vxzjs•t€n |q{é€ uv tv k€iéjz• = xwvxvijuq fsnlvkjzns•tv xyuuj wv éjoé¹ly nlqtq|
éjktj ( + + + + + ) = j xyuuj kxn} ·qxns éjktj
=
i Ívxuvzéqu rjrqn |q{é€ uv tv tn iéjz• Ívxrvs•ry xyuuj kxn} |q{é lnsqzx¹ tj zv tn iéjz• uv tv q vi¹ojzns•tv kunxzn qsq vltv qo ·qxns q vltv qo ·qxns dtj·qz uv tv iéjz• sq¡• ·qxsj q wvsy·jí•qnx¹ qo tq} wnénxzjtvkrjuq |q{é Çvsq·nxzkv ·qxns éjktv = fyuuj k
méywwn ·qxns wvsy·jí•q}x¹ wnénxzjtvkrjuq qo DEFG iylnz éjktj (D + E + F + G) vzxílj tj}vlqu vi•yí xyuuy
j Èylnu wqxjz• kunxzv insvmv ¡jéj q kunxzv ·nétvmv Ævo•unu tnrvzvévn éjoun•ntqn x wvkzvéntq¹uq qo tn ivsnn ·nu P
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Q nlqtq| cxsq vtv rvt·jnzx¹ tysnu zv lvwq¡nu k rvt|n |
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rvsq·nxzkn tnlvxzjí•nu lv |
Q + j oj tquq tysq lv |
P + cxsq |
n vtv rvt·jnzx¹ nlqtq|np zv lvwq¡nu k rvt|n tysq k |
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rvsq·nxzkn lv P + j oj tquq nlqtq|€ lv Q + Ívsy·qzx¹ wnén xzjtvkrj qo P + tysnp q Q + nlqtq| Ê viéjztv vziéjx€kj¹ qo rj lvp wnénxzjtvkrq qo P + tysnp q Q + nlqtq| wvxsnltqn lkn
méyww€ vlqtjrvk€} |q{é u€ wvsy·qu tnrvzvévn éjoun•ntqn x wv kzvéntq¹uq qo tn ivsnn ·nu P tysnp q Q nlqtq| Ën}vév¡quq ¹ks¹ ízx¹ sq¡• lkn wnénxzjtvkrq q ljí•qn wvxsn
ëzvp vwnéj|qq wyxzvn éjoun•ntqn Ìzxílj q xsnlynz ·zv ·qxsv zjrq} |
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éjoun•ntqp éjktv 3(P + Q + ) − |
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i ïqxsv éjoun•ntqp xvlné j•q} évktv |
N ins€} ¡jévk éjktv |
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3(N + |
Q + ) − 3(N Q + ) |
Ívëzvuy |
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∑ N>3(N + Q + ) − 3(N Q + )@ éjo Íénviéjoynu ëzv k€éj |
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N3(N + Q + ) − ∑ (N + )3(N + Q + ) = |
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= P3(P + Q + ) − ∑ 3(N + Q + ) = |
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= P3(P + Q + ) − 3(P Q + ) + = + PQ |
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Êxrvuvn ·qxsv éjktv xyuun ∑ |
∑ (S + T + )3(S T) Ëv |
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S= T= |
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(S + T + )3(S T) = (S + )∑ |
3(S T) + ∑ T3(S T) = |
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= (S + ) 3(S + Q) + ∑ 3(S + T − ) |
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T= |
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= (S + )>3(S + Q) + 3(S + Q − )@ = (S + )3(S + Q) |
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Ívëzvuy xyuuj éjktj |
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(S + )3(S + Q) = ∑ (S + )3(S + Q) − ∑ 3(S + Q) = |
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= (Q + )3(P + Q + ) − 3(P + Q + ) + = |
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= + PQ |
+ P + Q |
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P + Q + |
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