Y% &WordMicrosoft Word  r Courier New-@Times New Roman- )2  rSOMMES DE NOMBRES EN!/99(!+(+/9(+(!(+2  rTIERS. ((+! 2  r -)2  rPROGRESSIONS ARITHM%+/.+(!!/+!++(/9()2  rTIQUES ET GOMTRIQU(.+(!((.(/9((+.+2  rES(! 2  r @Times New Roman- 2 |b r -12 b rSommes de nombres entiers!00 !0 !- !b- 2 # r - 2 B rL"|2 B5K res sommes de nombres entiers sexpriment par des formules simples que nous  //////2 b  rrappelons ci 2  r-s2 E rdessous (les dmonstrations peuvent tre faites aussi par rcurrence)/ 2   r. 2   r @Symbol- 2 b r 2 ~ r >- 2  rS!2  romme S00!@Times New Roman- 2  rn-:2  r des n premiers nombres entiers ! 0!0 ! 2  r 2  r:  2  r r '- 2 Sb r S 2 S rS!@Times New Roman- 2 m rn-22 S r = 1 + 2 + 3 + 4 + + (n """""<" 2 S_ r2 S}  r 1) + n. " 2 S: r r '- 2 rS!- 2  rn-%2 4r = n ( n+ 1 ) / 2"" 2 r - !-- !-- !-- !-- !-- !a-- !-- !-- !-- !a-- !-- !--72 ?b rOn calcule en fait la somme S+//!-2 Y6 rn  - 2 YT r-2 Yh r 1,2  -2 ? r des n  2 ?? r@2 ?]# r 1 premiers carrs. On a videmment/+// 2 ? r  2 ? r: 2 ? r r '-2 nur(n  2 ur2 ur 1)- 2 ur2- 2 "ur 2 uBr= " 2 uBr -2 Br n- 2 Br2- 2 Br - 2 a~r  2 p~r2 ~r 2 n@Times New Roman-2 ~r x  - 2 ~r1 2 ,~r 2 }r+ 1" 2 )}r -2 nur(n  2 ur2 ur 2)- 2 ur2- 2 "ur 2 uBr= " 2 uBr -2 Br n- 2 Br2- 2 Br - 2 a~r  2 p~r2 ~r 2 n-2 ~r x  - 2 ~r2 2 ,~r -2 }r+ 2"- 2 }r2- 2 3}r -2 0nur(n  2 0ur2 0ur 3)- 2 &ur2- 2 0"ur 2 0uBr= " 2 0uBr -2 0Br n- 2 &Br2- 2 0Br - 2 0a~r  2 0p~r2 0~r 2 n-2 A~r x  - 2 0~r3 2 0,~r -2 0}r+ 3"- 2 &}r2- 2 &3}r - 2 kur -2 kuBr.(((( - 2 u0uBr -2 kSBr.(((( - 2 uBr  2 u~r  2 u}r -2 -ur(n  2 nur2 ur [n  2 ur2 ur 1])- 2 Our2- 2 cur 2 uBr= " 2 uBr -2 Br n- 2 Br2- 2 Br - 2  ~r  2 /~r2 M~r 2 n-2 ~r x  -2 ~r(n  2 ~r2 ,~r 1) 2 m~r -2 }r+ (n " 2 }r2 }r 1)- 2 `}r2- 2 t}r )2 "b rOn a une suite de n + 2 "@ r,2 "^ r 1 galits. On a donc+ 2 "g r  2 "v r: 2 " r r '- 2 rS!-2 rn  - 2 7r-2 Kr 1,2  - 2 r - 2 r=" 2 r -2 r(n  2 Or2 mr 1) n- 2 r2- 2 r  2 r2 r 2 n S!-2 rn  - 2 r-2 r 1 -2 r + S"!-2 Wrn  - 2 ur-2 r 1,2  - 2 r -2 r2 n S!-2 Urn  - 2 sr-2 r 1 - 2 r  2 r=" 2 r -2 r(n  2 :r2 Xr 1) n- 2 r2- 2 r - 2 rS!-2 .(rn  - 2 .Fr-2 .Zr 1 - 2 .xr - 2 r=" 2 r 2 r(n  2 &r2 Dr 1) n /2 2 r &2 zb ret enfin, en rempl/2 z  raant n  2 z r2 z r 1 par n 2 z r  2 z r: 2 z r r '- 2 ?rS!- 2 ?rn- 2 /?r - 2 S@r=" 2 u@r 2  rn (n + 1) /2" 2 r - 2 ' b r 2 ' ~ r >- 2 ,  rS!2 ,  romme S00!-2 G  rn,2 -J2 , * r des carrs des n premiers nombres entiers  ! 0!0 ! 2 ,  r 2 , # r:  2 , F r r '- 2 b r S 2  rS!-2  rn,2 -2  r = 1"- 2 f r2-2 z r + 2"- 2  r2-2  r + 3"- 2 J r2-2 ^ r + 4"- 2  r2-2   r + + (n "<" 2  r2  r 1)- 2 , r2-2 @ r + n"- 2  r2- 2  r  2  r r '- 2 rS!-2 ' rn,2 -.2 r = n (n+1) (2n + 1) / 6""" 2 r - ! -- ! -- ! -- ! -- ! -- !f -- !c -- !c -- !c -- !f -- !c -- !c -2 bU rLe calcul peut tre men comme prcdemment en exprimant la somme des n + 1 premiers "////////"/r '2 b rcubes 2  r 2  r:  2  r r '- 2 S 9r1- 2 I 9r3- 2 S 9r  2 S <r  2 S ;zr  2 S zr=" 2 S zr  2 S ,r - 2 I r - 2 I 9r - 2 I M9r - 2 I  r - 2 S "  d r1- 2 I @  d r3- 2 I T  d r - 2 9r2- 2 9r3- 2 9r  2 <r=" 2 ,<r -2 ;zr(1+1)"- 2 ;zr3- 2 );zr  2 zr=" 2 zr - 2 r1- 2 1r3- 2 Er  2 r+" 2 r - 2 9r3-2 9r x  - 2 29r1- 2 P9r2-2 d9r x  -2 9r1+" 2 9r - 2 M9r3-2 k9r x  - 2 9r1-2 9r x  - 2 9r1- 2 9r2- 2 9r  2  r+" 2  r - 2 "  d r1- 2 @  d r3- 2 T  d r  2 9r  2 <r  2 ;zr  2 zr  2 ,r  2 r  2 9r  2 M9r  2  r  2 ;  d r - 2 " 9rn- 2  9r3- 2 " 9r  2 " <r=" 2 " ,<r -2 " N;zr[(n  2 " ;zr2 " ;zr 1)+1]"- 2  V;zr3- 2 " j;zr  2 " zr=" 2 " zr -2 " r(n  2 " r2 " 1r 1)- 2  rr3- 2 " r  2 " r+" 2 " r - 2 " 9r3-2 3 9r x  -2 " 29r(n  2 " s9r2 " 9r 1)- 2  9r2-2 3 9r x  - 2 " 9r1 2 " *9r -2 " M9r+ 3"-2 3 9r x  -2 " 9r(n  2 " 9r2 " !9r 1)-2 3 a9r x  - 2 " 9r1- 2  9r2- 2 " 9r  2 "  r+" 2 "  r - 2 " "  d r1- 2  @  d r3- 2 " T  d r -2 h L9r(n+1)"- 2 ^ 9r3- 2 h 9r  2 h <r  2 h ;zr  2 h zr=" 2 h zr - 2 h rn- 2 ^ 1r3- 2 h Er  2 h r+" 2 h r - 2 h 9r3-2 y 9r x  - 2 h 29rn- 2 ^ P9r2-2 y d9r x  - 2 h 9r1 2 h 9r -2 h M9r+ 3"-2 y 9r x  - 2 h 9rn-2 y 9r x  - 2 h 9r1- 2 ^ $9r2- 2 h 89r  2 h  r+" 2 h  r - 2 h "  d r1- 2 ^ @  d r3- 2 h T  d r -[2 b5 rLa somme des membres de gauche donne la somme des n +"//////"(2  r 1 premiers cubes S/!-2  rn+1,3 - 2 3  r  2 B  r: 2 Q  r r '- 2 7 b r  2 7  rS!-2 Q , rn+1,3 -2 7  r = 1"- 2 -  r3-2 7  r + 2"- 2 - Y r3-2 7 m r + 3"- 2 -  r3-2 7  r + 4"- 2 - = r3-2 7 Q  r + + (n "<" 2 7 N r2 7 l r 1)- 2 -  r3-2 7  r + n"- 2 -  r3-2 7 3 r + (n+1)""- 2 -  r3-2 7  r = S"!-2 Q n rn,3 -2 7  r + (n+1)""- 2 - f  r3- 2 7 z  r r 'P2 b. ret la somme des membres de droite est gale //// 2  r r '- 2  rS!-2  rn,3 -2  r + 3"- 2 K r - 2 U rS!-2 v rn,2 -2  r + 3 S"!- 2 6 rn-2 J r + n + 1"" 2  r "2 Jb rLe calcul donne" 2 J r 2 J r:  2 J r r '- 2  rS!-2  rn,2 -/2  r = n (n+1) (2n + 1) / 6.""" 2   r r '-rr  qq  pp  oo  nn  mm  ll  kk  jj   i i    h h