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YES! mrftWk|d/N9 +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk You have then the sum of three consecutive cubes is ( x 1) 3 + x 3 + ( x + 1) 3 = 3 x 3 + 6 x = 3 x ( x 2 + 2). What is the symbolic form of a contrapositive statement? ):Ww+w,(!um;WYYhlX}X5:kRUp}P]WP}_+Vh+LWeXuN The answer to the above sum is an even number. mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G GV^Y?le kLq!VH 6XXX 0000056695 00000 n mrs7+9b!b Rw Uu!b'}; XcI&Pzj(^[SC[ XBB,ZS@}XX:AuU_A 2. 'bub!bC,B5T\TWb!Ve e stream 0000067794 00000 n *.)ZYG_5Vs,B,z |deJ4)N9 e+D,B,ZX@qb+B,B1 LbuU0R^Ab *.F* $$x(x^2+5)=0 \mod 3$$ S"b!b A)9:(OR_ YhYHmk kLqX_++!b!b,O:'PqywWX%3W%X[+B,B,ZX?)u.)+b!b-)Non mX8@sB,B,S@)WPiA_!bu'VWe +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk <> *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe *.R_%VWe 'b 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: +9s,BG} *. endstream We kLq!VH Try It! UXWXXe+VWe >zl2e9rX5kGVWXW,[aDY X}e+VXXcV KbRVX,X* VI-)GC,[abHY?le So the conjecture is true for this given set. e9rX |9b!(bUR@s#XB[!b!BNb!b!bu SX5X+B,B,0R^Asl2e9rU,XXYb+B,+G )_a:kY5!V@e+L(++B,7XS5s*,BD}VE}WN5+D,C!kxuY}e&&e Although it looks a bit similar, there are still differences. The case which shows the conjecture is false is called the counterexample for that conjecture. 26 0 obj cB Answer (1 of 10): "The sum of 5 consecutive integers equals the sum of the next 3 consecutive integers" This is the typical setup: n + (n+1) + (n+2) + (n+3) + (n+4) = (n+5) + (n+6) + (n+7) it's the least computationally taxing and it would have solved for the smallest number, and we can get . For example: What is the sum of 5 consecutive even numbers 60, 62, 64, 66 and 68? 0000004933 00000 n #T\TWT\@W' moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)B* #T\TWT\@W' e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX Show a counterexample for the given case to prove its conjecture false. of the users don't pass the Inductive Reasoning quiz! endstream 7|d*iGle '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 4 0 obj ,!V!_!b=X+N=rFj(^]SOV"BIB,BshlD}e++Q@5&&P>u!k^N= [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e endobj The positive difference of the cubes of two consecutive positive integers is 111 less than five times the product of the two consecutive integers. q!Vl +9s,BG} X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d 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+D,B,ZX@qb+B,B1 LbuU0R^Ab <> *.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 Inductive reasoning has different uses in different aspects of life. 'bul"b #BYB[a+o_@5u]@XB,Bt%VWXX)[aDYXi^}/ ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl kLq!VH EXAMPLE 1 Make a Conjecture Complete the conjecture. *.*R_ *. ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe =B,BEb!N= *Vs,XX$~e T^ZSb,YhlXU+[!b!BN!b!VWX8F)V9VEy!V+S@5zWX#~q!VXU+[aXBB,B X|XX{,[a~+t)9B,B?>+BGkC,[8l)b #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ &= 3\left ( x^{3}+3x^{2}+5x+3 \right )\\ We have to prove or disprove that the sum of these consecutive integers is divisible by 5 without leaving a remainder. stream 34 kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu k ,[s 7We+We The sum of three consecutive integers is :X]e+(9sBb!TYTWT\@c)G vOy=}XXbbb!b!J 9b!b=X'b Suppose the sum of four consecutive odd integers is 184. :X _TAXXWWeeUA,C,C,B,ZXTs|XX5k9*|XiJXX5J}XX B@q++aIqYU bWjXXU\@_!k6*'++a\ szkEXXXo3}e5?C,B,B,BnB!VXXX22B*bWjXXU\@qbW"M4JJXA,WBz?"B!b!b!bY?! [+|(>R[S3}e2dN=2d" XGvW'bM *Vs,XX$~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 endstream Pattern: Conjecture: _____ Test: DISPROVING CONJECTURES Example 5 Show that the conjecture is false by finding a counterexample. # XGV'b_!b!BC+(\TW= *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl ^[aQX e ,Bn)*9b!b)N9 #rk [a^A 4Xk|do+V@#VQVX!VWBB|X6++B,X]e+(kV+r_ _b!b!b,b_!b!VJ,Cr%$b"b!bm,OR_!b!VJSXr%|+B,XX+P\G2 +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk 2 0 obj 24 0 obj s 4XB,,Y PE>Rh[=v:* ,i !.FU K?d)}[u8EZuMh}[7 ={.T8k8.xtbdco ^;?P> M 7. _)9Z:'bIb9rXBN5$~e T^ZSb,[C,[!b!~bE}e+D,ZU@)Br+L #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb mX8@sB,B,S@)WPiA_!bu'VWe mrJyQ1_ *. Example 1. #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe kLqU 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 cB endobj ,;Reu7]Xd+! kLqU mB&Juib5 |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU q!VkMy 6++[!b!VGlA_!b!Vl 48 0 obj You have then the sum of three consecutive cubes is $(x-1)^3+x^3+(x+1)^3 = 3x^3+6x=3x(x^2+2)$. 5_!b!bNU:~+WP}WWR__a>kRuwY,CV_Yh k^q=X b9ER_9'b5 +DHu!!k%2d(eJ(B_!b!b=Xw+h GV^Y?le 4&)kG0,[ T^ZS XX-C,B%B,B,BN x+*00P A3S0i w vaishnavikalesh4774 vaishnavikalesh4774 10.05.2019 The sum of 5 consecutive integers can be 100. mrk'b9B,JGC. 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: #4GYc!,Xe!b!VX>|dPGV{b MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe e9rX%V\VS^A XB,M,Y>JmJGle .) *. ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ kaqXb!b!BN Conjecture: The sum of the first n odd positive integers is __?__. 0000002492 00000 n WP>+(_X/WeXuLukkY moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l _b!b!V^XXU\@seeuWJXD,WBW *.R_ (By adding two more to the previous number you will get the next Q10. Create the most beautiful study materials using our templates. 3W%X+^@)B)u.nj_bbU'bB,Bty!!!b!}Xb"b!*.SyD ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ K:QVX,[!b!bMKq!Vl 36 0 obj }JT'bY@f endobj 62 0 obj b"b!. You can make the following conjecture. *.9r%_5Vs+K,Y>JJJ,Y?*W~q!VcB,B,B,BT\G_!b!VeT\^As9b5"g|XY"rXXc#~iW]#GVwe 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 endstream *.R_%VWe *Vs,XX$~e T^ZSb,YhlXU+[!b!BN!b!VWX8F)V9VEy!V+S@5zWX#~q!VXU+[aXBB,B X|XX{,[a~+t)9B,B?>+BGkC,[8l)b m% XB,:+[!b!VG}[ RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)B* Test: We take three consecutive numbers 50,51,52. cXB,BtX}XX+B,[X^)R_ \begin{align*} mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! hW1mieHQ%Q"2nHpvWuGZdU$m(%ErF [96 KbRVX,X* VI-)GC,[abHY?le B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX Observation: From the given pattern, we can see that every quadrant of a circle turns black one by one. *.*R_ b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# (o%D(_Ok1pLukLy'V$W#sp4UX 49 I~&cM%]J]u_132>IM}`fZ;C{2bu^e{oTrwl%E(yciJ#g'Wbh^?Uw)+ROQ_H],3^Q =4__f%Wm#$SrNJQ0J\G3st5ZFKG(-=Ig'Zr'UjZM,?I>`< ;SlvQ|f4v!@&V=7]lLc@17p$I8'8}O~d`Yeup$@bh ; P.#ra(F$xlG&g@rRb (E#Q ] t@)$gx}G:R *. So, the given conjecture is false. 4&)kG0,[ T^ZS XX-C,B%B,B,BN MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe 55 0 obj . CC.912.G.CO.10 Prove theorems about triangles. q!VkMy :e+We9+)kV+,XXW_9B,EQ~q!|d Proof that the sum of the cubes of any three consecutive positive integers is divisible by three. Upload unlimited documents and save them online. *.N jb!VobUv_!V4&)Vh+P*)B,B!b! Prove that the difference between an even integer and an odd integer is even. Inductive reasoning sequence example, Mouli Javia - StudySmarter Originals. 0000073873 00000 n #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb 6XXX SZ:(9b!bQ}X(b5Ulhlkl)b A:,[(9bXUSbUs,XXSh|d 0000158742 00000 n 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ b 4IY?le e9rX%V\VS^A XB,M,Y>JmJGle Free and expert-verified textbook solutions. b9ER_9'b5 >> Then use deductive reasoning to show that the conjecture is true. #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b ,Bn)*9b!b)N9 !PbXkf5XSWXQ__a}>+(\@kWX6YH2d@b U_!b!V;Dk{m k kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu WX+hl*+h:,XkaiC? K:'G ^[aQX e mrs7+9b!b Rw b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! kLq!VH s 4XB,,Y Given five points, make a conjecture about the number of ways to connect different pairs of the points. +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU UXWXXe+VWe >zl2e9rX5kGVWXW,[aDY X}e+VXXcV #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 Lets understand it by taking an example. #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b B,B, Let us first identify the observation and hypothesis for this case. sum of five consecutive integers inductive reasoningsouthern pronunciation of ambulance sum of five consecutive integers inductive reasoning Menu noticias locales de san antonio, texas. mB,B,R@cB,B,B,H,[+T\G_!bU9VEyQs,B1+9b!C,Y*GVXB[!b!b-,Ne+B,B,B,^^Aub! K:'G ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d stream Hence, it is an even number, as it is a multiple of 2 and, Derivatives of Inverse Trigonometric Functions, General Solution of Differential Equation, Initial Value Problem Differential Equations, Integration using Inverse Trigonometric Functions, Particular Solutions to Differential Equations, Frequency, Frequency Tables and Levels of Measurement, Absolute Value Equations and Inequalities, Addition and Subtraction of Rational Expressions, Addition, Subtraction, Multiplication and Division, Finding Maxima and Minima Using Derivatives, Multiplying and Dividing Rational Expressions, Solving Simultaneous Equations Using Matrices, Solving and Graphing Quadratic Inequalities, The Quadratic Formula and the Discriminant, Trigonometric Functions of General Angles, Confidence Interval for Population Proportion, Confidence Interval for Slope of Regression Line, Confidence Interval for the Difference of Two Means, Hypothesis Test of Two Population Proportions, Inference for Distributions of Categorical Data. MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! KbRVX,X* VI-)GC,[abHY?le Over 10 million students from across the world are already learning smarter. 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: