Trying to understand how to get this basic Fourier Series. KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb 2 The product of three consecutive natural numbers can be equal to their sum. *. 'bu kLq!VH MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie #4GYcm }uZYcU(#B,Ye+'bu A number is a neat number if the sum of the cubes of its digit equals the number. #BI,WBW 5 weeks 6 days pregnant ultrasound pictures Projetos; is luke marrs adopted Blog; thomas aquinas natural law pdf Quem somos; . 0000074662 00000 n 16 0 obj 9b!b=X'b endstream 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: #Z:(9b!`bWPqq!Vk8*GVDY 4XW|#kG TYvW"B,B,BWebVQ9Vc9BIcGCSj,[aDYBB,ZF;B!b!b!b}(kEQVX,X59c!b!b'b}MY/ #XB[alXMl;B,B,B,z.*kE5X]e+(kV+R@sa_=c+hc!b! e_@s|X;jHTlBBql;B,B,B,Bc:+Zb!Vkb kLq!V>+B,BA Lb #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl _)9r_ <> e9rX%V\VS^A XB,M,Y>JmJGle U3}WR__a(+R@2d(zu!__!b=X%_!b!9 LbMU!R_Aj Although it looks a bit similar, there are still differences. 7 0 obj "T\TWbe+VWe9rXU+XXh|d*)M|de+'bu 2d++Lu_+(\@5(C!k6YYTmmR_!b!b!b!be+L0A e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX 18 0 obj *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe GY~~2d}WO !N=2d" XGv*kxu!R_Ap7j(nU__a(>R[SOjY X,CV:nb!b!b! +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk .)ZbEe+V(9s,z__WyP]WPqq!s,B,,Y+W+MIZe+(Vh+D,5u]@X2B,ZRBB,Bx=UYo"ET+[a89b!b=XGQ(GBYB[a_ :X]e+(9sBb!TYTWT\@c)G >> #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, 8 0 obj mrJyQ1_ *. 6XXX |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb #4GYcm }uZYcU(#B,Ye+'bu 'b 7|d*iGle ,!V!_!b=X+N=rFj(^]SOV"BIB,BshlD}e++Q@5&&P>u!k^N= XbbbUn++W5USbB,B,*.OB!lb)UN,WBW 7|d*iGle 3 0 obj mrftWk|d/N9 KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb !*beXXMBl <> Example #4: Look at the following patterns: 3 -4 = -12 To To prove that a conjecture is true, you need to prove it is true in all cases. 'bu ~+t)9B,BtWkRq!VXR@b}W>lE 6++[!b!VGlA_!b!Vl mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs kByQ9VEyUq!|+E,XX54KkYqU WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)B* 0000117474 00000 n b>X+B,XX+P\D2 *. 0000053628 00000 n =*GVDY 4XB*VX,B,B,jb|XXXK+ho Let the first number be n #n+(n+1)=5# simplified to #2n=4# divide by 2 gives #n=2 and (n+1)=3# Answer link . moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l ^[aQX e ,B2dT'b}Yg4XCe(&}XGX5X, |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s +R@Y/eZ,C X,BBBI*f,BD}Q_!bEj(^[S!C2d(zu!!++B,::kRJ}+l)0Q_A{WX Y!@YhY~Xi_!b!9 X2dU+(\TW_aKY~~ :X]e+(9sBb!TYTWT\@c)G m%e+,RVX,B,B)B,B,B LbuU0+B"b 59 0 obj b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! k~u!AuU_Ajj,*VX=N :>6'b9d9dEj(^[S n+Vzu!|J KVX!VB,B5$VWe m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> mB&Juib5 s 4Xc!b!F*b!TY>" *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* XXX22B,}r%t%XXU\@se^_!b'VRr%t% +!b!V+B,B,bg~%SXXb!V*eeX!}JJCO cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X This is a high school question though, so if someone can explain it to me in a highschool math language, it will be appreciated. 'Db}WXX8kiyWX"Qe We&+(\]S$!\"b:e&P#}5Xw*kKu=X Find the smallest number. ,B,HmM9d} b9duhlHu!"BI!b!1+B,X}QVp}P]U' bVeXXOTV@z!>_UCCC,[!b!bV_!b!b!bN|}P]WP}X(VX=N :}5X*rr&Pk(}^@5)B,:[}XXXSe+|AuU_AnPb,[0Q_A{;b!1z!|XC,,[a65pb}*VXQb!b!B#WXXie endobj endobj _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& N=2d" Yu!>+BB,ZT@uh}2dY_A{WWp}P]U'b} Y mX+#B8+ j,[eiXb mrAU+XBF!pb5UlW>b 4IYB[aJ}XX+bEWXe+V9s Suppose x and y are odd integers. So, doves and geese are both of the same species. B,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb *.F* # XGV'b_!b!BC+(\TW= mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G ,B,HiMYZSbhlB XiVU)VXXSV'30 *jQ@)[a+~XiMVJyQs,B,S@5uM\S8G4Kk8k~:,[!b!bM)N ZY@O#wB,B,BNT\TWT\^AYC_5V0R^As9b!*/.K_!b!V\YiMjT@5u]@ bW]uRY XB,B% XB,B,BNT\TWT\^Aue+|(9s,B) T^C_5Vb!bkHJK8V'}X'e+_@se+D,B1 Xw|XXX}e S 42 B. mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G *.F* The positive difference of the cubes of two consecutive positive integers is 111 less than five times the product of the two consecutive integers. X>+kG0,[!b}X!*!b |X+B,B,,[aZ)=zle9rU,B,%|8g TY=?*W~q5!{}4&)Vh+D,B} XbqR^AYeE|X+F~+tQs,BJKy'b5 acknowledge that you have read and understood our, Data Structure & Algorithm Classes (Live), Data Structure & Algorithm-Self Paced(C++/JAVA), Android App Development with Kotlin(Live), Full Stack Development with React & Node JS(Live), GATE CS Original Papers and Official Keys, ISRO CS Original Papers and Official Keys, ISRO CS Syllabus for Scientist/Engineer Exam, Number expressed as sum of five consecutive integers, Represent the fraction of two numbers in the string format, Check if a given array contains duplicate elements within k distance from each other, Find duplicates in a given array when elements are not limited to a range, Find duplicates in O(n) time and O(1) extra space | Set 1, Find the two repeating elements in a given array, Duplicates in an array in O(n) and by using O(1) extra space | Set-2, Duplicates in an array in O(n) time and by using O(1) extra space | Set-3, Count frequencies of all elements in array in O(1) extra space and O(n) time, Find the frequency of a number in an array, Count number of occurrences (or frequency) in a sorted array, Merge two sorted arrays with O(1) extra space, Efficiently merging two sorted arrays with O(1) extra space, Program for Nth node from the end of a Linked List, Write a function that counts the number of times a given int occurs in a Linked List, Add two numbers represented by Linked List, Add two numbers represented by linked lists | Set 2, Add two numbers represented by Linked List without any extra space, Tree Traversals (Inorder, Preorder and Postorder). _QAXX5l#22!b!b *9B,B,T@seeXU[b)UN,WBW <> b 4IY?le B,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb *. cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ Prove that the difference between an even integer and an odd integer is even. +e+|V+MIB,B,B}T+B,X^YB[aEy/-lAU,X'Sc!buG ~+t)9B,BtWkRq!VXR@b}W>lE ,Bn)*9b!b)N9 |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb :X]e+(9sBb!TYTWT\@c)G j XYYuu!b}lXB,BCe_!b=XSe+WP>+(\_A*_ b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B ,X'PyiMm+B,+G*/*/N }_ k VXT9\ ] +JX=_!,9*!m_!+B,C,C OyQ9VE}XGe+V(9s,B,Z9_!b!bjT@se+#}WYlBB,jbM"KqRVXA_!e Conjecture Number 20 must be divisible by 5. |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb 30 0 obj *Vs,XX$~e T^ZSb,YhlXU+[!b!BN!b!VWX8F)V9VEy!V+S@5zWX#~q!VXU+[aXBB,B X|XX{,[a~+t)9B,B?>+BGkC,[8l)b *. MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 *. X>+kG0,[!b}X!*!b |X+B,B,,[aZ)=zle9rU,B,%|8g TY=?*W~q5!{}4&)Vh+D,B} XbqR^AYeE|X+F~+tQs,BJKy'b5 'bub!bC,B5T\TWb!Ve &!t_j IYY~XbMXjf5XSWXQ__a}>+(\@kWX6YHUMM:~+D,jXUwbM@bMU_aEY~~pu!_!b2d"+CV66)!b-#VN5kV5UY~e&:W X~ejetY,BBvXu/!AY $TeVWWp_} _)9Z:'bIb9rXBN5$~e T^ZSb,[C,[!b!~bE}e+D,ZU@)Br+L 0000003474 00000 n endobj mrAU+XBF!pb5UlW>b 4IYB[aJ}XX+bEWXe+V9s kV)!R_A{5WXT'b&WXzu!!(C4b U!5X~XWXXuWX=+ZC,B 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ mX8@sB,B,S@)WPiA_!bu'VWe ,B,HiMYZSbhlB XiVU)VXXSV'30 *jQ@)[a+~XiMVJyQs,B,S@5uM\S8G4Kk8k~:,[!b!bM)N ZY@O#wB,B,BNT\TWT\^AYC_5V0R^As9b!*/.K_!b!V\YiMjT@5u]@ bW]uRY XB,B% XB,B,BNT\TWT\^Aue+|(9s,B) T^C_5Vb!bkHJK8V'}X'e+_@se+D,B1 Xw|XXX}e +9Vc}Xq- moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l Let us consider two integer numbers say -2 and -3. Here, the statements are true, but the conjecture made from it is false. e We *.R_ m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L To make a conjecture, we first find a pattern. k X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d Since the middle integer, n, . kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu 20 C. 12 D. 30 E. 56 16. . cEV'PmM UYJK}uX>|d'b As $3x(x^2+2)$ will have a multiple of three occurring once in the $3$, and once in either the $x$ or the $(x^2+2)$ term, we have that the sum of three consecutive cubes is a multiple of nine. kLq!V e+D,B,ZX@qb+B,B1 LbuU0R^Ab +9s,BG} Everything you need for your studies in one place. *.vq_ *.N jb!VobUv_!V4&)Vh+P*)B,B!b! #rk [a^A 4Xk|do+V@#VQVX!VWBB|X6++B,X]e+(kV+r_ >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ mB&Juib5 m |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb endobj mX8kSHyQV0n*Qs,B,/ XB,M,YC[aR>Zle Where possible, show work to support your conclusion. *|eeU+C,B,zb!b!Vqy!!!}_!+a\ ] +JXXS|XXX+g\ ] K|eXX8SbbUWXXH_5%V/,B,BC,C,CB,W"bV mrJyQszN9s,B,ZY@s#V^_%VSe(Vh+PQzlX'bujVb!bkHF+hc#VWm9b!C,YG eFe+_@1JVXyq!Vf+-+B,jQObuU0R^As+fU l*+]@s#+6b!0eV(Vx8S}UlBB,W@JS +e+|V+MIB,B,B}T+B,X^YB[aEy/-lAU,X'Sc!buG 4GYc}Wl*9b!U +9s,BG} 0000151388 00000 n Conjecture: The next number will be 16, because 11+5=16. $VRr%t% +Wb(jb!bC@}e*12B,B,Zv_!b!VJ,CjPUiJK&kc}XXz+MrbV+b5 +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe 9b!b=X'b mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs mX+#B8+ j,[eiXb K:'G Now we just have to prove $3|x$ or $3|x^2+2$. bbb!b=XiDXXXh^Jk9*'++a\ +'B,B,B/_UV'buvB22 !!b!~b +!b!b!C,CrbX"VRr%t% +!b!DbX!B,ZR?s|JW%2B,B,ZY@^B)22 !!b!Nb&+!b!b!C,CbX%VRr%t% +!b!bX-B,ZR?s|JW%2B,B,ZY@+m$H,C,C cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X endobj mrftWk|d/N9 ~iJ[WXX2B,BA X;_!b!VijJ,W\ kNy}XXBN!b/MsqUWXX58knb!bh*_5%+aXX5HB,Bxq++aIi ~+^@)B)u.nj_bbU'bB,Bty!!!b!}Xb"b!*.Sy, ,[s _)9Z:'bIb9rXBN5$~e T^ZSb,[C,[!b!~bE}e+D,ZU@)Br+L k <> WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d !PbXkf5XSWXQ__a}>+(\@kWX6YH2d@b U_!b!V;Dk{m k s 4XB,,Y "bU@5)BD}P]5WXe+|(Vh+LT'b,rr&P+,^@5)B, !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!VWe kLqU mrs7+9b!b Rw Formula for sum of 'n' terms of an arithmetic sequence: S n = n 2 [ 2 a 1 + ( n - 1) d]. 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! mrJy!VA:9s,BGkC,[gFQ_eU,[BYXXi!b!b!b!b')+m!B'Vh+ sW+hc}Xi s,XX8GJ+#+,[BYBB8,[!b!b!BN#??XB,j,[(9]_})N1: s,Bty!B,W,[aDY X: :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e k~u!B,[v_!bm= [ b65CVKi_9d9dN="b!^J *Vs,XX$~e T^ZSb,YhlXU+[!b!BN!b!VWX8F)V9VEy!V+S@5zWX#~q!VXU+[aXBB,B X|XX{,[a~+t)9B,B?>+BGkC,[8l)b endobj [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e Consider groups of three consecutive numbers. #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe 35 hW1mieHQ%Q"2nHpvWuGZdU$m(%ErF [96 7WWXQ__a(Y7WSe2dMW!C,BBe_!b!b!CV_A +C,C!++C!&!N b|XXXWe+B #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe SInce there are 5 integers, 5x5 would make the sum 25 greater than A. 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD Xg&PJ,CV:e&PvE_!b!b!#M`eV+h mB,B,R@cB,B,B,H,[+T\G_!bU9VEyQs,B1+9b!C,Y*GVXB[!b!b-,Ne+B,B,B,^^Aub! x+*00P A3S0ih ~ 0000068633 00000 n Get 247 customer support help when you place a homework help service order with us. W'3ezWuB,C!B&XXT'P>+(:X, Best study tips and tricks for your exams. So, about 70% of doves are white. Like even numbers, odd numbers are integers that are not divisible by 2. X>+kG0,[!b}X!*!b |X+B,B,,[aZ)=zle9rU,B,%|8g TY=?*W~q5!{}4&)Vh+D,B} XbqR^AYeE|X+F~+tQs,BJKy'b5 Do}XXXXKJ,Ckaq=X?b!b!Vqy!!!b$_$++a\ kNyWXX3W%Xo mrJyQszN9s,B,ZY@s#V^_%VSe(Vh+PQzlX'bujVb!bkHF+hc#VWm9b!C,YG eFe+_@1JVXyq!Vf+-+B,jQObuU0R^As+fU l*+]@s#+6b!0eV(Vx8S}UlBB,W@JS For example: What is the sum of 5 consecutive even numbers 60, 62, 64, 66 and 68? stream !MU'b [aN>+kG0,[!b!b!>_!b!b!V++XX]e+(9sB}R@c)GCVb+GBYB[!b!bXB,BtXO!MeXXse+V9+4GYo%VH.N1r8}[aZG5XM#+,[BYXs,B,B,W@WXXe+tUQ^AsU{GC,X*+^@sUb!bUA,[v+m,[!b!b!z8B,Bf!lbuU0R^Asu+C,[s #4GYc!,Xe!b!VX>|dPGV{b K:QVX,[!b!bMKq!Vl 'b mX+#B8+ j,[eiXb :X mX+#B8+ j,[eiXb *.N1rV'b5GVDYB[aoiV} T^ZS T^@e+D,B,oQQpVVQs,XXU- mrJyQ1_ )#j(^[S MxmM]W'FN b!bR@zg_ Conjecture: The sum of five consecutive integers is always divisible by five. nb!Vwb WbY[SY_:_Yu!!MxmM]&P:k Answer (1 of 4): Any five consecutive integers can be expressed as n-2, n-1, n, n+1 and n+2, for some n (the middle integer of the five). of the users don't pass the Inductive Reasoning quiz! _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b mrs7+9b!b Rw w,[0Q_AN O buj(^[S=d >_9d9dhlBB5 4X?+B,B,::AuU_A 4GYc}Wl*9b!U Integers are three types of numbers including negative integers, positive integers and zero. +++LtU}h #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl cXB,BtX}XX+B,[X^)R_ #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b Step 3 Test your conjecture using other numbers. XF+4GYkc!b5(O9e+,)M.nj_=#VQ~q!VKb!b:X +9Vc}Xq- B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX #rk [a^A 4Xk|do+V@#VQVX!VWBB|X6++B,X]e+(kV+r_ <> k^q=X _)9r_ cB +M,[; [+|(!!kY X,CV65XWX&X WX+hl*+h:,XkaiC? #4GYcm }uZYcU(#B,Ye+'bu UyA +GY~E_WWX5 XY,CV_YY~5:H_!b!bRC_a(k._N5++LYCCVT ,C!k6 mX8kSHyQV0n*Qs,B,/ XB,M,YC[aR>Zle qWX5 B:~+TW~-b&WN}!|e5!5X,CV:A}XXBJ}QC_a>+l0A,BeTUW,CxbYBI!Cb!b *GY~~_aX~~ b"VX,CV}e2d'!N b=X_+B,bU+h b e+D,B1 X:+B,B,bE+ho|XU,[s e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e 4GYc}Wl*9b!U x mq]wEuIID\\EwL|4A|^qf9r__/Or?S??QwB,KJK4Kk8F4~8*Wb!b!b+nAB,Bxq! mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe x+*00P A3S0ih ~* cEV'PmM UYJK}uX>|d'b b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B ZXW~keq!F_!bXXXXS|JJ+)BJSXr%D+N)B,B,B,qqU+aQo_b!b!b,N +B"bbbUk\ ] a!b!b'b5bX5XiJXXq>!b!bC,j^?s|JgV'bmb!V*eeXO'VZM(Ir%D,B,X@sbXXiJXXq2!b!b mrk'b9B,JGC. *.N jb!VobUv_!V4&)Vh+P*)B,B!b! wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e stream #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! We&+(\]SufmMe[}5X+N=2d" W'b_!b!B,CjY}+h kLq!V 9 0 obj *Vs,XX$~e T^ZSb,YhlXU+[!b!BN!b!VWX8F)V9VEy!V+S@5zWX#~q!VXU+[aXBB,B X|XX{,[a~+t)9B,B?>+BGkC,[8l)b b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! 0dfjWP(0Q_Az&Y!:_Yu!!MxmM]W'bMB,B,R@$AuL_ _)9Z:'bIb9rXBN5$~e T^ZSb,[C,[!b!~bE}e+D,ZU@)Br+L Think of it this way, each of the next 5 consecutive positive integers is 5 more than the corresponding first five integers. 3. Let n is sum of five consecutive integer of k 2, k-1, k, k + 1, k+2. 4GYc}Wl*9b!U Yes I got it now. *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe [aN>+kG0,[!b!b!>_!b!b!V++XX]e+(9sB}R@c)GCVb+GBYB[!b!bXB,BtXO!MeXXse+V9+4GYo%VH.N1r8}[aZG5XM#+,[BYXs,B,B,W@WXXe+tUQ^AsU{GC,X*+^@sUb!bUA,[v+m,[!b!b!z8B,Bf!lbuU0R^Asu+C,[s *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 UyA kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! mB,B,R@cB,B,B,H,[+T\G_!bU9VEyQs,B1+9b!C,Y*GVXB[!b!b-,Ne+B,B,B,^^Aub! Therefore, the sum of 5 consecutive odd numbers is equal to 5 times the third odd number. kLq!V stream +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk S 54 0 obj #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ $$x(x^2+5)=0 \mod 3$$ kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! Proof: x = 3 k x 0 ( mod 3) +DHu!!kU!@Y,CVBY~Xg+B,XGY~#~mYO,B #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, 'bu WP>+(_X/WeXuLukkY ~iJ;WXX2B,BA X}+B,J'bbb!bUSbFJXXsNAub!b)9r%t%,)j? 0000003548 00000 n *.R_%VWe mrJy!VA:9s,BGkC,[gFQ_eU,[BYXXi!b!b!b!b')+m!B'Vh+ sW+hc}Xi s,XX8GJ+#+,[BYBB8,[!b!b!BN#??XB,j,[(9]_})N1: s,Bty!B,W,[aDY X: [5_bn~3;D+dlL._L>; ,S=& endstream endobj 365 0 obj <>stream m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> sum of five consecutive integers inductive reasoning. kByQ9VEyUq!|+E,XX54KkYqU VX>+kG0oGV4KhlXX{WXX)M|XUV@ce+tUA,XXY_}yyUq!b!Vz~d5Um#+S@e+"b!V>o_@QXVb!be+V9s,+Q5XM#+[9_=X>2 4IYB[a+o_@QXB,B,,[s +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk 0000073513 00000 n kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b!
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