:e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e 9b!b=X'b *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 ,Bn)*9b!b)N9 #Z: Example 2: The sum of an odd and an even number If an odd number and an even number are added, will the sum be an odd or an even number? How to Sum Integers 1 to n. You dont need to be a math whiz to be a good programmer, but there are a handful of equations you will want to add to your problem solving toolbox. A place where magic is studied and practiced? 'b S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu 'b So not all predicted conclusions can be true. 4&)kG0,[ T^ZS XX-C,B%B,B,BN Therefore, the sum of 5 consecutive integers can be expressed by the formula: N + (N + 1) + (N + 2) + (N + 3) + (N + 4). +e+|V+MIB,B,B}T+B,X^YB[aEy/-lAU,X'Sc!buG The sum of five consecutive integers is divisible by 5 is indeed true; for if we denote the five consecutive integers by n, then n . UyA [+|(>R[S3}e2dN=2d" XGvW'bM B,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)B* A:,[(9bXUSbUs,XXSh|d 3W%X+^@)B)u.nj_bbU'bB,Bty!!!b!}Xb"b!*.SyD [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e endstream +C,C!++C!&!N b|XXXWe+B ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ mX+#B8+ j,[eiXb *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 'bub!bC,B5T\TWb!Ve ?oWP>+(\@5(C!k6YYTmmR_!b!b!>+B,W __aX~Wp}P]WP:kP,ClbY _}wmkkuj5TYX 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b kByQ9VEyUq!|+E,XX54KkYqU WX+hl*+h:,XkaiC? X~~ b"V:e^eY,Ce"b!VWXXO$! mrftWk|d/N9 _~WXXX)B,@w b9ER_9'b5 mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe Example: Prove the sum of two odd numbers is an even number. Let the consecutive numbers be n and n + 1. SX5X+B,B,0R^Asl2e9rU,XXYb+B,+G 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 mX8@sB,B,S@)WPiA_!bu'VWe ?*'++a\ nsB,B,BN!VWO:XX_!bXXXX#|JJAC/ SX5X+B,B,0R^Asl2e9rU,XXYb+B,+G 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l kLqU Therefore, the sum of 5 consecutive odd numbers is, (2 * N + 1) + (2 * N + 3) + (2 * N + 5) + (2 * N + 7) + (2 * N + 9), = 2 * N + 1 + 2 * N + 3 + 2 * N + 5 + 2 * N + 7 + 2 * N + 9. endstream w0dV+h :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e _WX B,B,22 !!b!b-6'bbb &VWmT9\ ] +JXXsZ+B,jbg\ ] KZ+B,jb!b!bmUbbbUWXXh+JSXr%D,B9-b!b53W%b!b5**eeXX+B,B 4XXXb)UN,WBW 1. 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: #4GYcm }uZYcU(#B,Ye+'bu +^u!_!b2d"+CV66)!bNkB5UY~e&:W~ZC,B2de2dE:WZmmRC_!b!V;:Xu_!b!k Uu!b'}; XcI&Pzj(^[SC[ XBB,ZS@}XX:AuU_A *.)ZYG_5Vs,B,z |deJ4)N9 kLq!V #T\TWT\@W' EXAMPLE 1 Make a Conjecture Complete the conjecture. SR^AsT'b&PyiM]'uWl:XXK;WX:X mrJy!VA:9s,BGkC,[gFQ_eU,[BYXXi!b!b!b!b')+m!B'Vh+ sW+hc}Xi s,XX8GJ+#+,[BYBB8,[!b!b!BN#??XB,j,[(9]_})N1: s,Bty!B,W,[aDY X: s 4Xc!b!F*b!TY>" #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ q!Vl ,X'PyiMm+B,+G*/*/N }_ Inductive reasoning consists of the following steps: Observe the sample set and identify the patterns. WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX q!VkMy p}P]WPAuUOQ_ *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 m%e+,RVX,B,B)B,B,B LbuU0+B"b B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX <> (b) Write 1346 as the sum of four consecutive integers. ,BB:X+C!k~u!!MxuM!b!BI!VAuU_AdE,w+h WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d Need to show that #BYB[a+o_@5u]@XB,Bt%VWXX)[aDYXi^}/ MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe e MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie e+D,B,ZX@qb+B,B1 LbuU0R^Ab ,|B,ZB,_@{MxmM]W'IVRT'bB,_@e+&+(\TWp_ kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu ?l k SR^AsT'b&PyiM]'uWl:XXK;WX:X +e+|V+MIB,B,B}T+B,X^YB[aEy/-lAU,X'Sc!buG #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ mrftWk|d/N9 #Z: YhYHmk q!V[22B,B X+[+B 4XXXXc+W mB&Juib5 %PDF-1.7 % For example: What is the sum of 5 consecutive integers 15, 16, 17, 18 and 19? *.F* _)9r_ The use process is also very simple, input the first integer, then select the integer type. S"b!b A)9:(OR_ +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk #T\TWT\@W' *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe 0000054170 00000 n #4GYc!,Xe!b!VX>|dPGV{b *.*R_ let a and b be odd numbers. Therefore,k-2 + k-1 + k + k+1 + k+2= n5*k = nThe five numbers will be n/5 2, n/5 1, n/5, n/5 + 1, n/5 + 2. 'bub!bCHyUyWPqyP]WTyQs,XXSuWX4Kk4V+N9"b!BNB,BxXAuU^AT\TWb+ho" X+GVc!bIJK4k8|#+V@se+D,B1 X|XXB,[+U^Ase+tUQ^A5X+krXXJK4Kk+N9 +++L'bi&WV@fj*Y2d^@{WXU&O~OXX[hY~ b 'b endstream bXJXX+z_bgVWX+B,C,C@jiJK&kc}XXz+MrbV:BXB,BthB3WXXX++B,W]e!!!F:OyiL"+!b!b! UY~~ e"VX,CV|5WY,ClbYBI!V}XXXs+h 6++[!b!VGlA_!b!Vl |d/N9 How do I find the angles of an isosceles triangle whose two base angles are equal and whose third angle is 10 less than three times a base angle? 16 0 obj 7|d*iGle n = number of 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 KbRVX,X* VI-)GC,[abHY?le 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ *.)ZYG_5Vs,B,z |deJ4)N9 +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU What can you say about the sum of any two odd integers? ,Bn)*9b!b)N9 0000094336 00000 n _,9rkLib!V |d*)M.N B}W:XXKu_!b!b** *.N jb!VobUv_!V4&)Vh+P*)B,B!b! #BYB[a+o_@5u]@XB,Bt%VWXX)[aDYXi^}/ q!VkMy >S?s|JJXR?B,B,B,W?)u.o*kaq!WX.O922B,m_5%+aXX5BB,Bxq++aIi ~+B,'bu ?l 'b WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b 41 0 obj .)ZbEe+V(9s,z__WyP]WPqq!s,B,,Y+W+MIZe+(Vh+D,5u]@X2B,ZRBB,Bx=UYo"ET+[a89b!b=XGQ(GBYB[a_ *.vq_ Therefore, 153 is a neat number. ~+t)9B,BtWkRq!VXR@b}W>lE 0000066998 00000 n *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 *.N jb!VobUv_!V4&)Vh+P*)B,B!b! Using the formula to calculate, the third odd integer is 85, so its 5 times is 5 * 85= 425. It is sufficient to show only one counterexample to prove the conjecture false. 43 0 obj A conjecture is said to be true if it is true for all the cases and observations. GV^Y?le 203 0 obj << /Linearized 1 /O 210 /H [ 3548 1385 ] /L 484577 /E 187344 /N 6 /T 480398 >> endobj xref 203 107 0000000016 00000 n cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X Be perfectly prepared on time with an individual plan. _,9rkLib!V |d*)M.N B}W:XXKu_!b!b** 'Db}WXX8kiyWX"Qe >> k^q=X =*GVDY 4XB*VX,B,B,jb|XXXK+ho You use inductive reasoning when you find a pattern in specific cases and then write a conjecture for the general case. cEV'PmM UYJK}uX>|d'b b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# :X]e+(9sBb!TYTWT\@c)G From the above, we can observe that the answer of all the sums is always an even number. 0000152179 00000 n Find 3 Consecutive Even Integers with a Sum of 72 Consecutive integers can be found by starting with an integer n and adding one to it repeatedly to form a sequence. >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ mX+#B8+ j,[eiXb A:,[(9bXUSbUs,XXSh|d mX8@sB,B,S@)WPiA_!bu'VWe As we all know, even numbers are integers divisible by 2. 6XXX _)9Z:'bIb9rXBN5$~e T^ZSb,[C,[!b!~bE}e+D,ZU@)Br+L Prove that the negative of any even integer is even. 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ 34 +9Vc}Xq- nb!Vwb 4&)kG0,[ T^ZS XX-C,B%B,B,BN 7|d*iGle 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b Proof: $x=3k\Rightarrow x\equiv 0\pmod{3}$, $x=3k\pm 1\Rightarrow x^2 \equiv (\pm 1)^2 \equiv 1\pmod{3}\Rightarrow x^2+2\equiv 0\pmod{3}$. KJkeqM=X+[!b!b *N ZY@b!b! ,Bn)*9b!b)N9 kaqXb!b!BN mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe q!Vl W/?o *R_A{WWNg_ X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: 18 0 obj Answer link. e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e K:'G Contrapositive: If a number is not a whole number, then i is not a natural number 0000055164 00000 n #4GYc!,Xe!b!VX>|dPGV{b 0000068151 00000 n 6XXX !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!VWe stream endobj 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ #Z: We m Number 20 ends with 0. 0000117497 00000 n 7|d*iGle 0000144950 00000 n e Inductive Reasoning is the process of reasoning to a general conclusion through . ~iJ[+C,C JSXw%XXXXX/6j5UWbbEe!V@4S^?JXXWXX$VRr%t% +k|!b!b!b!b!}b5u*O+C,B,B%D,Bx }XXbb=eJ_=XiJK&kI4JJXAWC,B,B,B,z4z:'Pqq!b!b!F_"b!VJ,C6Rz:OyiL"+!b!b!>_!b!b=XiJXY_=`XXXX#VW?k_ +^C_u%!VXXXi[OyiJK&k~@,B,z$*'++a_ X+KXXB~ T\^S*.12B,B,WBB,W]e!!!VSOyiJK&_h 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe UXWXXe+VWe >zl2e9rX5kGVWXW,[aDY X}e+VXXcV mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe UyA For example, since $4 \times 2 = 8$, the probability of landing on 8 . mrftWk|d/N9 e9rX |9b!(bUR@s#XB[!b!BNb!b!bu Does ZnSO4 + H2 at high pressure reverses to Zn + H2SO4? _,9rkLib!V |d*)M.N B}W:XXKu_!b!b** |d/N9 SX5X+B,B,0R^Asl2e9rU,XXYb+B,+G The case which shows the conjecture is false is called a counterexample for that conjecture. W+,XX58kA=TY>" kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu ,XF++[aXc!VS _Y}XTY>"/N9"0beU@,[!b!b)N b!VUX)We Its 100% free. ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ # XGV'b_!b!BC+(\TW= *.*b +9Vc}Xq- True/False: What is the answer to the conjecture? m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L =*GVDY 4XB*VX,B,B,jb|XXXK+ho #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe ?l 65 0 obj ,Bn)*9b!b)N9 Suppose x and y are odd integers. moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l Andy made 4 more stars per minute than Belen. e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e SZ:(9b!bQ}X(b5Ulhlkl)b k ,X'PyiMm+B,+G*/*/N }_ 9b!b=X'b 6XXX mX8kSHyQV0n*Qs,B,/ XB,M,YC[aR>Zle Inductive reasoning allows the prediction of future outcomes. MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe 0000057246 00000 n However, when using inductive reasoning, even though the statement is true, the conclusion wont necessarily be true. +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe *. #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e _N b!\b}b!b!BI!V+BlD}QXc!VX,N=rr&P|"VXXV'Xb] cB 'bu B,B, 0000072355 00000 n mX+#B8+ j,[eiXb G Uu!zu@,C!UMxmM=tj(^]S$_]zBI!b!1 _)9Z:'bIb9rXBN5$~e T^ZSb,[C,[!b!~bE}e+D,ZU@)Br+L %PDF-1.4 % MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie The meaning of the questions: given n, n can be written in the form of at least two consecutive positive integers and the number of species. mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G Using the formula to calculate, the third integer is 17, so its 5 times is 5 * 17 = 85. In this tutorial, you learned how to sum a series of consecutive integers with a simple and easy to remember equation. rev2023.3.3.43278. #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl Are inductive and deductive the same type of reasoning? 34 XF+4GYkc!b5(O9e+,)M.nj_=#VQ~q!VKb!b:X ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ =*GVDY 4XB*VX,B,B,jb|XXXK+ho B,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb "T\TWbe+VWe9rXU+XXh|d*)M|de+'bu This gives us our starting point. <> The sum of 5 consecutive positive integers = A. K:'G [+|(!!kY X,CV65XWX&X $$x(x^2+5)=0 \mod 3$$ KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! m% XB,:+[!b!VG}[ 2. *. Make a test a conjecture about the sum of any three consecutive integers. "l!O)|jn17,JwO@$ p,z(f`D0UH i4#6a #7n4f2 E$"94%8~\Ygtp9Y>qhtj8grgb{FjxAaQ{n=Gko +lHb. #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b _b!b!B6B,BM 4XXXXr%V'PqyM+B,S?s|JJXR?WX8SXKSz_bbU'bb!bm*O922B,*+aWXb!+WBWAVB,B= XB,_RWXX58kSy!!!b=Xr%V'PqyM+B,S?s|JJXR?WX8SXKSz_bbU'bb!bm*O922Br%V'PqyM+B,S?s|JJXR?WX8SXKSz_bbU'bb!bm*O922BG++W\ ] keyB,B=3W%X|XX{:Xu4!!VkPq!V_!b!C,C,C,ZKXX5b!+WBWAVB,B= XB,_RWXX58L4kqy!!!b"VZSr%t% +!b!b)O:WXJ,N)B,+OyqM}XXbbb!b!z~+B,BC,C,C,OI,WBW :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e 71 0 obj ,X'PyiMm+B,+G*/*/N }_ b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! S K:QVX,[!b!bMKq!Vl 34 endobj d+We9rX/V"s,X.O TCbWVEBj,Ye Inductive reasoning is considered to be predictive rather than certain. +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe x+*00P A3S0i w@ moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l 0000172339 00000 n +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ !*beXXMBl 0000127093 00000 n #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, Conjecture is the general conclusion which we reached by using induction reasoning. 39 0 obj k^q=X 'bub!bC,B5T\TWb!Ve Get. Test: We take three consecutive numbers 50,51,52. 7. *. Truth value: false; 0 ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B V_keq!V++2!!VjJ_XXX 4XXXBJSXr%D,Bb_!b!b!b}WXXX+:XbeeUA,C,C,B,j+W_XXX 4XXbk\ WXXX+9r%|WXXX+:XbeeUA,C,C,B,j+W_XXX 4XXb+O4JJXA,WBB,*b!b!b!g\ u%|V'bu N +B,:(Vh+LWP>+[aKYoc!b!&P~Wc5TYYYhlXBI!b%B,[a(V;V:kn}PXX]b9d9dEj(^[SC ^@5)B, *.N jb!VobUv_!V4&)Vh+P*)B,B!b! &!t_j IYY~XbMXjf5XSWXQ__a}>+(\@kWX6YHUMM:~+D,jXUwbM@bMU_aEY~~pu!_!b2d"+CV66)!b-#VN5kV5UY~e&:W X~ejetY,BBvXu/!AY $TeVWWp_} ~+t)9B,BtWkRq!VXR@b}W>lE ,X'PyiMm+B,+G*/*/N }_ 7WWX=++LT'bY@fj*YC,C!+R@N C_#;5UY~ |d/N9 |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s If yes, find the five consecutive integers, else print -1.Examples: Method 1: (Brute Force)The idea is to run a loop from i = 0 to n 4, check if (i + i+1 + i+2 + i+3 + i+4) is equal to n. Also, check if n is positive or negative and accordingly increment or decrement i by 1.Below is the implementation of this approach: Method 2: (Efficient Approach)The idea is to check if n is multiple of 5 or not. 17 0 obj 16060 ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b 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 mX+#B8+ j,[eiXb _)9r_ 44 0 obj 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! Let us understand it by taking an example. 55 0 obj WX+hl*+h:,XkaiC? mrJyQ1_ MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe wl|k^Mx rr,hlX_ k~u!l m%e+,RVX,B,B)B,B,B LbuU0+B"b Upload unlimited documents and save them online. cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X Multiple Choice Which of the following is a counterexample of the conjecture below? *.N1rV'b5GVDYB[aoiV} T^ZS T^@e+D,B,oQQpVVQs,XXU- e9rX%V\VS^A XB,M,Y>JmJGle wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b ?l {3W}}eXX8S#beeUA,C,C,B,j+W_XXX 4XXX9_!xb)UN,WBW 0000144927 00000 n ^[aQX e +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk [aN>+kG0,[!b!b!>_!b!b!V++XX]e+(9sB}R@c)GCVb+GBYB[!b!bXB,BtXO!MeXXse+V9+4GYo%VH.N1r8}[aZG5XM#+,[BYXs,B,B,W@WXXe+tUQ^AsU{GC,X*+^@sUb!bUA,[v+m,[!b!b!z8B,Bf!lbuU0R^Asu+C,[s 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b SZ:(9b!bQ}X(b5Ulhlkl)b endstream 'bu *. e+D,B,ZX@qb+B,B1 LbuU0R^Ab mX+#B8+ j,[eiXb +DHu!!k!@Y,CVBY~Xb!b!ez(p0+ #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl /:X*0,BBee2de2dE&X_!b!b!GY~~0D,B *.)ZYG_5Vs,B,z |deJ4)N9 *.vq_ *. RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)B* e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX e wV__a(>R[S3}e2dN=2d" XGvW'bM endobj 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: For example, if you leave for work and it's raining outside, you reasonably assume that it will rain the whole way and decide to carry an umbrella. A. #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 6++[!b!VGlA_!b!Vl *.R_ Using the formula to calculate, the third even integer is 64, so its 5 times is 5 * 64 = 320, the answer is correct. e 0000053628 00000 n ~WXUYc9(O j1_9rU,B,58[!_=X'#VX,[tWBB,BV!b=X uWX'VXA,XWe%q_=c+tQs,B58kVX+#+,[BYXUXWXXe+tUQ^AsWBXerkLq! _TAXX+uWXX5 >> stream S"b!b A)9:(OR_ 'bub!bC,B5T\TWb!Ve cEZ:Ps,XX$~eb!V{bUR@se+D/M\S 'b K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU 2 The product of three consecutive natural numbers can be equal to their sum. mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs mX8kSHyQV0n*Qs,B,/ XB,M,YC[aR>Zle Proof: x = 3 k x 0 ( mod 3) #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb The conclusions obtained via inductive reasoning are only probable but not certain. So the numbers are 18, 19, 20, 21, 22 and statement 1 is correct. mrs7+9b!b Rw #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 Prove that the difference between an even integer and an odd integer is even. 'b log(x1)log(x+2)=log(x+2)log(x1). kLqU cEZ:Ps,XX$~eb!V{bUR@se+D/M\S 1 5, 1 6, 1 7, 1 8, 1 9. 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 mrAU+XBF!pb5UlW>b 4IYB[aJ}XX+bEWXe+V9s 0000058374 00000 n 'b which shows that n is sum of ve consecutive integers. ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl |d/N9 mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab 6Xb}kkq!B,B,T?)u.)/MsqU'b,N w|X)O922B,S@5W e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e 13 0 obj ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl 6XXX SR^AsT'b&PyiM]'uWl:XXK;WX:X b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# [GYXr:+Zu!VN ::kb!bS_AjU_A{e+&+(\TW XikBuCYmkkrU'b 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ #4GYcm }uZYcU(#B,Ye+'bu 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: 0000065974 00000 n endobj *. XXXSXXX22B,BUSbB,B,*.O922jJbMMbVtWXXB,B!b!b!}bbbUvWMNBI,WBW x -qo@"EyCv?Oc?/?='rvx`??j; b 4IY?le l|X which marvel character matches your personality. What is an advantage of using inductive reasoning? cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ <> 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 Below is the implementation of this approach: Find last five digits of a given five digit number raised to power five, Count numbers up to N that cannot be expressed as sum of at least two consecutive positive integers, Check if a number can be expressed as a sum of consecutive numbers, Count primes that can be expressed as sum of two consecutive primes and 1, Count prime numbers that can be expressed as sum of consecutive prime numbers, Check if a given number can be expressed as pair-sum of sum of first X natural numbers, Check if a number can be expressed as sum two abundant numbers, Check if a number can be expressed as sum of two Perfect powers, Check if a number N can be expressed as the sum of powers of X or not, Check if a prime number can be expressed as sum of two Prime Numbers. +9s,BG} mrs7+9b!b Rw According to inductive reasoning, the sum of two negative integers is always negative. +e+|V+MIB,B,B}T+B,X^YB[aEy/-lAU,X'Sc!buG kaqXb!b!BN Converse: If a number is a whole number, then it is a natural number #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk
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