"T\TWbe+VWe9rXU+XXh|d*)M|de+'bu In addition to manual calculations, this page also provides a calculator for calculating the sum of 5 consecutive integers, so that you can get the sum of 5 consecutive integers faster. RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)B* ?*'++a\ B Just because all the people you happen to have met from a town were strange is no guarantee that all the people there are strange. b 4IY?le k~u!B,[v_!bm= e+D,B1 X:+B,B,bE+ho|XU,[s WX+hl*+h:,XkaiC? The smaller of two consecutive integers is eight less than A straightforward word problem solved using an equation. Upload unlimited documents and save them online. S"b!b A)9:(OR_ vOy=}XXbbb!b!J *.N1rV'b5GVDYB[aoiV} T^ZS T^@e+D,B,oQQpVVQs,XXU- 0000107763 00000 n [+|(>R[S3}e2dN=2d" XGvW'bM <> #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe 7|d*iGle =*GVDY 4XB*VX,B,B,jb|XXXK+ho e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX s 4Xc!b!F*b!TY>" mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs :e+We9+)kV+,XXW_9B,EQ~q!|d XXX|uXXX22B,Bb!b!C,C,C%z+MrbVWX XXX22B,E}JJB,O4JJXA,WBBjb}WXX) You use inductive reasoning when you find a pattern in specific cases and then write a conjecture for the general case. !*beXXMBl Each sum is even. endobj In inductive reasoning, true observations might have false conjecture. Make a test a conjecture about the sum of any three consecutive integers. 'bul"b Using the formula to calculate, the third integer is 17, so its 5 times is 5 * 17 = 85. q!Vl The difference between an even integer and an odd integer is odd. 0000074662 00000 n 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 :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e R22 !!b!b5+/,B,BC,CC_!xb)UN,WBW by John Pegg and Angel Gutierrez. stream kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu mrAU+XBF!pb5UlW>b 4IYB[aJ}XX+bEWXe+V9s For $$x=\pm 1 \mod 3$$, *.R_%VWe Suppose x and y are odd integers. x -qo@"EyCv?Oc?/?='rvx`??j; SZ:(9b!bQ}X(b5Ulhlkl)b Since the middle integer, n, . endstream +DHu!!kU!@Y,CVBY~Xg+B,XGY~#~mYO,B ,[s ):bKU'bYumkBXO!!k}P]5WcGY~~ MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe +"b!Bu+B,W'*e Using the formula to calculate, the third even integer is 64, so its 5 times is 5 * 64 = 320, the answer is correct. 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: *.9r%_5Vs+K,Y>JJJ,Y?*W~q!VcB,B,B,BT\G_!b!VeT\^As9b5"g|XY"rXXc#~iW]#GVwe "T\TWbe+VWe9rXU+XXh|d*)M|de+'bu *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD *.*R_ q!VkMy cB ,BB:X+C!k~u!!MxuM!b!BI!VAuU_AdE,w+h 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 *.N jb!VobUv_!V4&)Vh+P*)B,B!b! 'b b 4IY?le 0000161836 00000 n 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: e+D,B,ZX@qb+B,B1 LbuU0R^Ab *.vq_ +9s,BG} 60 + 62 + 64 + 66 + 68 = 320. 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 !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!VWe endstream UXWXXe+VWe >zl2e9rX5kGVWXW,[aDY X}e+VXXcV MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie b *.)ZYG_5Vs,B,z |deJ4)N9 endobj SR^AsT'b&PyiM]'uWl:XXK;WX:X GV^Y?le +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU However, inductive reasoning does play a part in the discovery of mathematical truths. *.*b So, the formula for the sum of 5 consecutive even numbers is, 2 * N + (2 * N + 2) + (2 * N + 4) + (2 * N + 6) + (2 * N + 8), = 2 * N + 2 * N + 2 + 2 * N + 4 + 2 * N + 6 + 2 * N + 8. >> _)9r_ UXWXXe+VWe >zl2e9rX5kGVWXW,[aDY X}e+VXXcV _)9r_ S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu stream Case $2: x=3k+1$, then $x^2+2=9k^2+6k+1+2=3(3k^2+2k+1)$. x+*00P A3S0i w 16060 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 KbRVX,X* VI-)GC,[abHY?le KJkeqM=X+[!b!b *N ZY@b!b! Also, to prove the newly formed conjecture true in all similar circumstances, we need to test it for other similar evidence. ,XF++[aXc!VS _Y}XTY>"/N9"0beU@,[!b!b)N b!VUX)We endobj endobj =*GVDY 4XB*VX,B,B,jb|XXXK+ho s 4XB,,Y *. _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b fairbanks ice dogs standings . <> *.F* 9b!b=X'b 0000053428 00000 n kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! b #T\TWT\@W' *.9r%_5Vs+K,Y>JJJ,Y?*W~q!VcB,B,B,BT\G_!b!VeT\^As9b5"g|XY"rXXc#~iW]#GVwe Let S be the number of perfect squares among the integers from 1 to 20136. +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb UyA #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 *.R_ A:,[(9bXUSbUs,XXSh|d s 4Xc!b!F*b!TY>" Sign up to highlight and take notes. mrk'b9B,JGC. cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X X2dU+(\TWu__aX~We"V65u;}e2d X,BB+B,W'bMUp}P]RW~~!bS_A{WX9C[2dYC,C_!b!_!b!V:kRJ}++ K:'G S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu !*beXXMBl e For example, test that it works with . Third, click calculate button to get the answer. 9b!b=X'b m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> stream mrJyQ1_ 'bul"b mrJy!VA:9s,BGkC,[gFQ_eU,[BYXXi!b!b!b!b')+m!B'Vh+ sW+hc}Xi s,XX8GJ+#+,[BYBB8,[!b!b!BN#??XB,j,[(9]_})N1: s,Bty!B,W,[aDY X: RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)B* d+We9rX/V"s,X.O TCbWVEBj,Ye VX>+kG0oGV4KhlXX{WXX)M|XUV@ce+tUA,XXY_}yyUq!b!Vz~d5Um#+S@e+"b!V>o_@QXVb!be+V9s,+Q5XM#+[9_=X>2 4IYB[a+o_@QXB,B,,[s |d/N9 Trying to understand how to get this basic Fourier Series. 'Db}WXX8kiyWX"Qe mrJyQszN9s,B,ZY@s#V^_%VSe(Vh+PQzlX'bujVb!bkHF+hc#VWm9b!C,YG eFe+_@1JVXyq!Vf+-+B,jQObuU0R^As+fU l*+]@s#+6b!0eV(Vx8S}UlBB,W@JS :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ m%e+,RVX,B,B)B,B,B LbuU0+B"b ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ The sum of 5 consecutive integers is equal to 5 times the third integer. SZ:(9b!bQ}X(b5Ulhlkl)b WX+hl*+h:,XkaiC? S: s,B,T\MB,B5$~e 4XB[a_ 16060 *Vs,XX$~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 ?+B,XyQ9Vk::,XHJKsz|d*)N9"b!N'bu |d/N9 *.)ZYG_5Vs,B,z |deJ4)N9 Where does this (supposedly) Gibson quote come from? <> e +++Wp}P]WP:YmbY _e Z If a number is a natural number, then it is also a whole number, Inverse: IF a number is not a natural number, then it is not a whole number % Uu!:vC,C!+R@z&PC__!b!b-N :AuU_DQ_=++LWP>$QCC,C!+R@z&P&U'bZ_AYoWe&+(\TW XGk;}XoU'bYC65u^_!b!b-N :AuU_JQ_=++LWP>>[[SYo kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab KJkeqM=X+[!b!b *N ZY@b!b! kaqXb!b!BN . x 2 + y 2 Archimedes, Newton, Gauss If the conjecture is FALSE, give a counterexample. Short story taking place on a toroidal planet or moon involving flying. 'bub!bC,B5T\TWb!Ve [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 Case $3: x=3k+2$, then $x^2+2=9k^2+12+4+2=3(3k^2+4k+2)$. 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e #Z: Choose the correct conjecture for the following? *.*b !*beXXMBl mrs7+9b!b Rw ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ 7We+We d+We9rX/V"s,X.O TCbWVEBj,Ye ~WXUYc9(O j1_9rU,B,58[!_=X'#VX,[tWBB,BV!b=X uWX'VXA,XWe%q_=c+tQs,B58kVX+#+,[BYXUXWXXe+tUQ^AsWBXerkLq! ,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 B,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb $$x(x^2+5)=0 \mod 3$$ 0000144927 00000 n b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! kLq!V>+B,BA Lb UyA This reasoning gives a chance to explore the hypothesis in a wider field. No need to think about the whole process. 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: Contrapositive: If a number is not a whole number, then i is not a natural number SX5X+B,B,0R^Asl2e9rU,XXYb+B,+G 55 0 obj If the statement is false, make the necessary change(s) to produce a true statement. 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ *.R_ VX>+kG0oGV4KhlXX{WXX)M|XUV@ce+tUA,XXY_}yyUq!b!Vz~d5Um#+S@e+"b!V>o_@QXVb!be+V9s,+Q5XM#+[9_=X>2 4IYB[a+o_@QXB,B,,[s B,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb 6++[!b!VGlA_!b!Vl B,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb XW+b!5u]@K 4X>l% T^\Syq!Bb!b ** XF+4GYkc!b5(O9e+,)M.nj_=#VQ~q!VKb!b:X mX8@sB,B,S@)WPiA_!bu'VWe |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb *. Deductive reasoning is a reasoning method that makes conclusions based on multiple logical premises which are known to be true. e+D,B1 X:+B,B,bE+ho|XU,[s hV0}nIan4PMB=F>ZxilSIVLYm5I4AN1%4bA A{auu'QNlx3xLc'/?O+ovWmxm<30Sdwurnx9VfzNzzV 9%^sJvvzfhalO6&#JAJT~Hwi526.,S;=A5xmJT,*r6IErPa~D}"A;P0F C3DBB,,Q8]@K #Z:(9b!`bWPqq!Vk8*GVDY 4XW|#kG TYvW"B,B,BWebVQ9Vc9BIcGCSj,[aDYBB,ZF;B!b!b!b}(kEQVX,X59c!b!b'b}MY/ #XB[alXMl;B,B,B,z.*kE5X]e+(kV+R@sa_=c+hc!b! e_@s|X;jHTlBBql;B,B,B,Bc:+Zb!Vkb +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk Math. 'bu endstream #T\TWT\@W' d+We9rX/V"s,X.O TCbWVEBj,Ye To prove this conjecture true for all even numbers, lets take a general example for all even numbers. <> moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l _)9r_ w b 4IY?le endobj 1 Answer Phillip E. May 31, 2015 2 and 3. *.R_%VWe endobj mB&Juib5 Let us understand it by taking an example. Formula for sum of 'n' terms of an arithmetic sequence: S n = n 2 [ 2 a 1 + ( n - 1) d]. *Vs,XX$~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 KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb endobj KJkeqM=X+[!b!b *N ZY@b!b! b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! ,[s 34 Consider 2 and 5. K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& Hypothesis: Both numbers taken must be positive. 0000008844 00000 n e9rX%V\VS^A XB,M,Y>JmJGle endstream MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie x+*00P A3S0i w[ #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl 59 0 obj So about 70% of doves in the U.S. are white. <> What is the symbolic form of a contrapositive statement? GV^Y?le e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e mB#V_!W'bZ_Ap7|b KVX!VB,B5$VWe X2dU+(\TW__aX~We"V65oW,C!^@{e+D,Z+B,W'bMUp}P]Fb&WN}Q_!bEj(^[S;o{e2d X,BBBI*_aKY~~ K:'G b 4IY?le =W~GWXQ_!bYkh~SY!kYe"b!Fb}WuDXe+L 20 0 obj Is a PhD visitor considered as a visiting scholar? What can you say about the sum of any two odd integers? x mq]wEuIID\\EwL|4A|^qf9r__/Or?S??QwB,KJK4Kk8F4~8*Wb!b!b+nAB,Bxq! I can prove deductively that they are divisible by $3$ but so far any combination I choose fails to prove the divisibility by $9$. mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX So $n = 21$.} 65 0 obj |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s _,9rkLib!V |d*)M.N B}W:XXKu_!b!b** b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# Does ZnSO4 + H2 at high pressure reverses to Zn + H2SO4? mrs7+9b!b Rw e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX mrftWk|d/N9 0000136972 00000 n # XGV'bkBXuL}B,,[0Q_AN BOp}!f|e u#}UN="b!BIB,BzXp}+hlc%NxmM}b!|b9d9dEj(^[S N +[a:kRXuHu!$_!b!V=WP>+(\_Ajl Find the counterexample to prove this conjecture false. *.*b "T\TWbe+VWe9rXU+XXh|d*)M|de+'bu |d/N9 e9rX%V\VS^A XB,M,Y>JmJGle S: s,B,T\MB,B5$~e 4XB[a_ nb!Vwb So, we can use 2 * N + 1 to represent the first integer, then the remaining 3 consecutive odd numbers can be represented as 2 * N + 3, 2 * N + 5, 2 * N + 7 and 2 * N + 9. #4GYcm }uZYcU(#B,Ye+'bu moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l Example #4: Look at the following patterns: 3 -4 = -12 The sum of 5 consecu 1. *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe UyA mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs >> *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab endstream s 4Xc!b!F*b!TY>" e9rX%V\VS^A XB,M,Y>JmJGle *. 4GYc}Wl*9b!U b9ER_9'b5 m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L e9rX%V\VS^A XB,M,Y>JmJGle 7 0 obj If there is no solution, output -1. X+WBW +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU kaqXb!b!BN 9b!b=X'b We 0000072332 00000 n K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& 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. Hence, the smallest number is 43. nb!Vwb True statement My dog is brown. #rk [a^A 4Xk|do+V@#VQVX!VWBB|X6++B,X]e+(kV+r_ 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? >> +e+|V+MIB,B,B}T+B,X^YB[aEy/-lAU,X'Sc!buG x+*00P A3S0i wR Xg&PJ,CV:e&PvE_!b!b!#M`eV+h Inductive reasoning is often used to predict future outcomes.
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