+9_aX~~ bS@5:_Yu}e2d'!N=+D,k@XuWXO e9rX |9b!(bUR@s#XB[!b!BNb!b!bu *Vs,XX$~e T^ZSb,YhlXU+[!b!BN!b!VWX8F)V9VEy!V+S@5zWX#~q!VXU+[aXBB,B X|XX{,[a~+t)9B,B?>+BGkC,[8l)b A:,[(9bXUSbUs,XXSh|d b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# Here the difference between two numbers 2 and 3 is greater than its sum. 35 B. *.N jb!VobUv_!V4&)Vh+P*)B,B!b! hW1mieHQ%Q"2nHpvWuGZdU$m(%ErF [96 =*GVDY 4XB*VX,B,B,jb|XXXK+ho *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! e9rX |9b!(bUR@s#XB[!b!BNb!b!bu #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb k~u!AuU_A4"_;GY~~z&Ya_YhYHmk 1. +9Vc}Xq- CC.912.G.CO.11 Prove theorems about parallelograms. UXWXXe+VWe >zl2e9rX5kGVWXW,[aDY X}e+VXXcV +|>kRujJeO,C!+R@{WX&}XXB,,J}>E}W"__aX~'bMj WV]Pi_Ye2dEh *.*b _)9r_ 34 mrk'b9B,JGC. 6++[!b!VGlA_!b!Vl kLq!V>+B,BA Lb A question of NUMBER THEORY and divisibility of 7. K:QVX,[!b!bMKq!Vl U}S*+ U}WCu 0000152257 00000 n m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu ?l +++LtU}h The different types of inductive reasonings are categorized as follows: This form of reasoning gives the conclusion of a broader population from a small sample. WX+hl*+h:,XkaiC? A:,[(9bXUSbUs,XXSh|d Notice that the sum of the five . k^q=X *.N1rV'b5GVDYB[aoiV} T^ZS T^@e+D,B,oQQpVVQs,XXU- 'Db}WXX8kiyWX"Qe +e+|V+MIB,B,B}T+B,X^YB[aEy/-lAU,X'Sc!buG 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). 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 wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U Step 1 1 of 3. *. :e+We9+)kV+,XXW_9B,EQ~q!|d That is, the sum of 5 consecutive even numbers is equal to 5 times the third even number. +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe Make a conjecture about a given pattern and find the next one in the sequence. s 4XB,,Y ~+t)9B,BtWkRq!VXR@b}W>lE *.R_ "T\TWbe+VWe9rXU+XXh|d*)M|de+'bu )#j(^[S MxmM]W'FN b!bR@zg_ * 42 0 obj *.*R_ Let $X$ stand for any natural number and let $X+1$ and $X+2$ stand for the two consecutive numbers. _b!b!V^XXU\@seeuWJXD,WBW Using the formula to calculate, the third integer is 17, so its 5 times is 5 * 17 = 85. *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* KVX!VB,B5$VWe X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe 16060 |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s endobj #rk [a^A 4Xk|do+V@#VQVX!VWBB|X6++B,X]e+(kV+r_ e9rX |9b!(bUR@s#XB[!b!BNb!b!bu Therefore, 153 is a neat number. We have to prove or disprove that the sum of these consecutive integers is divisible by 5 without leaving a remainder. #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e Nie wieder prokastinieren mit unseren Lernerinnerungen. q!Vl Therefore, let the first even number is 2 * N, then the remaining 4 consecutive even numbers can be expressed as 2 * N + 2, 2 * N + 4, 2 * N + 6and 2 * N + 8. Sign up to highlight and take notes. :e+We9+)kV+,XXW_9B,EQ~q!|d 11 31 3 51 3 5 7 1 12 4 22 9 32 16 42 ANSWER The sum of the first n . _!!b&!0A,w+hn_VWX,CC({|e:,CVEY~Xu*~WuDXe+L B,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb p}P]WPAuUOQ_ 'bub!bC,B5T\TWb!Ve B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX K:'G XW+b!5u]@K 4X>l% T^\Syq!Bb!b ** kLq!VH ,Bn)*9b!b)N9 :XW22B,BN!b!_!bXXXXS|JJkWXT9\ ] +JXb!b!bu mB&Juib5 kLqU Will you pass the quiz? In math, what does reciprocal mean? UyA S: s,B,T\MB,B5$~e 4XB[a_ *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD ,|WR__aBY~N=2dn!@Y,CVBY~Xb!b!ez(p0+ EXAMPLE 1. Prove that a group of even order must have an element of order 2. "T\TWbe+VWe9rXU+XXh|d*)M|de+'bu N bU+(\TWbe+&+h|N|B,::!!+R@nZ mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs Numbers 3, 4, and 5 are called consecutive integers. kaqXb!b!BN Consider some even numbers, say, 68, 102. 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 ,XF++[aXc!VS _Y}XTY>"/N9"0beU@,[!b!b)N b!VUX)We 2 The product of three consecutive natural numbers can be equal to their sum. Case $2: x=3k+1$, then $x^2+2=9k^2+6k+1+2=3(3k^2+2k+1)$. So not all predicted conclusions can be true. +9Vc}Xq- 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. Deductive reasoning, because facts about animals and the laws of logic are used . XXXMl#22!b!b *n9B,B,T@seePb}WmT9\ ] +JXXsWX kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu N b!bR@uF+B,VN}(Vf}QXX)3kkC!C,,[a:B}WXXp}P]RWX1e e+D,B1 X:+B,B,bE+ho|XU,[s b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B S *.*R_ mB,B,R@cB,B,B,H,[+T\G_!bU9VEyQs,B1+9b!C,Y*GVXB[!b!b-,Ne+B,B,B,^^Aub! 16060 OyQ9VE}XGe+V(9s,B,Z9_!b!bjT@se+#}WYlBB,jbM"KqRVXA_!e |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD 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 b 18 0 obj !*beXXMBl Find the next 5 terms in the sequence 38, 31, 24, 17, ___, ___, ___, ___, ___ . S: s,B,T\MB,B5$~e 4XB[a_ stream +M,[; What sort of strategies would a medieval military use against a fantasy giant? If you preorder a special airline meal (e.g. |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb l|X \end{align*}, This can be used to deductively prove that the sum of cube of $3$ consecutive numbers is divisible by $3$ but I can't prove it is divisible by $9$. We & (x)^{3}+(x+1)^{3}+(x+2)^{3}\\ |d/N9 KJkeqM=X+[!b!b *N ZY@b!b! We&+(\]SufmMe[}5X+N=2d" W'b_!b!B,CjY}+h +9s,BG} Top Questions. s 4Xc!b!F*b!TY>" =++DceZ+C!k~u!!MxuM!nb!BI!VAuU_AE,w+h Free and expert-verified textbook solutions. k^q=X KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb mX8kSHyQV0n*Qs,B,/ XB,M,YC[aR>Zle mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe '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 mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs #4GYcm }uZYcU(#B,Ye+'bu *. RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)B* ANSWER The sum of any two odd numbers is even. S: s,B,T\MB,B5$~e 4XB[a_ 7WWXQ__a(Y7WSe2dMW!C,BBzWXXu$*kWPM`eVWW=B,CV6TbYez:k(>+B,B,:XS5s+(\_A&j W+,XX58kA=TY>" SZ:(9b!bQ}X(b5Ulhlkl)b mrs7+9b!b Rw ,[s By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. 21 0 obj 0000055055 00000 n The case which shows the conjecture is false is called a counterexample for that conjecture. 0000071968 00000 n 'bu 'Db}WXX8kiyWX"Qe knXX5vOy=}XXbbb!b!D To prove this conjecture true for all even numbers, lets take a general example for all even numbers. K:'G Then use deductive reasoning to show that the conjecture is true. W/?o *R_A{WWNg_ K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& ,X'PyiMm+B,+G*/*/N }_ *.vq_ mrJyQ1_ In addition to calculating the sum of 5 consecutive integers, you may also need to calculate the sum of 5 consecutive even numbers, or the sum of 5 consecutive odd numbers. Observation: From the given pattern, we can see that every quadrant of a circle turns black one by one. *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* S #4GYc!,Xe!b!VX>|dPGV{b - The product of two odd numbers is odd. b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B 52 0 obj 16060 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe mB&Juib5 *.N1rV'b5GVDYB[aoiV} T^ZS T^@e+D,B,oQQpVVQs,XXU- [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e mrftWk|d/N9 Step 1: Find a rule by using few examples. 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ ,[s B,B,R@B,B,BI endobj I need to deductively prove that the sum of cubes of $3$ consecutive natural numbers is divisible by $9$. kLq!V You can make the following conjecture. *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD *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 Is a PhD visitor considered as a visiting scholar? Which of the above statements is/are correct ? N=2d" Yu!>+BB,ZT@uh}2dY_A{WWp}P]U'b} Y mrs7+9b!b Rw kLqU K:'G cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ endobj endobj k~u!AuU_Abe+|(Vh+LT'b6'b9d9dEj(^[S x_9de+|:kRXuH !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!VWe 19 0 obj e+D,B1 X:+B,B,bE+ho|XU,[s Now we just have to prove $3|x$ or $3|x^2+2$. Find two consecutive even integers whose sum is 126. #rk [a^A 4Xk|do+V@#VQVX!VWBB|X6++B,X]e+(kV+r_ mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab ,Bn)*9b!b)N9 |d/N9 Therefore, the sum of 5 consecutive odd numbers is equal to 5 times the third odd number. kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B stream S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu '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: kLqU endstream W+,XX58kA=TY>" *. *. |d/N9 'bu XF+4GYkc!b5(O9e+,)M.nj_=#VQ~q!VKb!b:X cEV'PmM UYJK}uX>|d'b U}E}b,[0Q_A{;XX|B,P@{MxmM]WRWO8d d+We9rX/V"s,X.O TCbWVEBj,Ye Yes I got it now. mrk'b9B,JGC. 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$H,C,C |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb +9Vc}Xq- If there is no solution, output -1. m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! #4GYcm }uZYcU(#B,Ye+'bu e+D,B,ZX@qb+B,B1 LbuU0R^Ab 0000055164 00000 n U'bY@uduS-b!b p}P]WPAuU_A/GYoc!bS@r+rr^@Mxu![ XB,BCS_Ap}:%VK=#5ufmM=WYb9d kLq!V What is the sum of the first 20 Z? wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ Answer by ikleyn(44793) ( Show Source ): Also, the sum of first 'n' positive integers can be calculated as, Sum of first n positive integers = n (n + 1)/2, where n is the total number of integers. kLqU ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ (a) Prove: If n is the sum of 4 consecutive integers, then n is not divisible by 4. 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KJkeqM=X+[!b!b *N ZY@b!b! 0000004910 00000 n RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)B* !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!VWe SR^AsT'b&PyiM]'uWl:XXK;WX:X 0000057246 00000 n ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B _)9r_ a) Describe two different algorithms for finding a spanning tree in a simple graph. 9b!b=X'b mrftWk|d/N9 _)9r_ s 4XB,,Y ,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 endobj b"b! 4GYc}Wl*9b!U *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD ~+t)9B,BtWkRq!VXR@b}W>lE !*beXXMBl *.9r%_5Vs+K,Y>JJJ,Y?*W~q!VcB,B,B,BT\G_!b!VeT\^As9b5"g|XY"rXXc#~iW]#GVwe #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 *. We mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"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: b"b!:*b!b53W%uT+B,jb!b!b =+C,C 4 0 obj x+*00P A3S0i wR #rk [a^A 4Xk|do+V@#VQVX!VWBB|X6++B,X]e+(kV+r_ !*beXXMBl N R_Ajl-e K:QVX,[!b!bMKq!Vl +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU mrJyQ1_ 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 For building our understanding of the world, inductive reasoning is used in day-to-day life. 7|d*iGle sum of five consecutive integers inductive reasoningsouthern pronunciation of ambulance sum of five consecutive integers inductive reasoning Menu noticias locales de san antonio, texas. cEZ:Ps,XX$~eb!V{bUR@se+D/M\S m%e+,RVX,B,B)B,B,B LbuU0+B"b Inductive reasoning vs. Deductive reasoning, slideplayer.com. . 25 0 obj B,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb x+*00P A3S0i wd q!VkMy mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs _,9rkLib!V |d*)M.N B}W:XXKu_!b!b** Step 4: Test conjecture for all even numbers. #Z: [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e UN=!;khY,CVX~X"B,!5X~Weuh b 4IY?le mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ #4GYcm }uZYcU(#B,Ye+'bu ^[aQX e #4GYc!,Xe!b!VX>|dPGV{b _)9Z:'bIb9rXBN5$~e T^ZSb,[C,[!b!~bE}e+D,ZU@)Br+L Determine whether each equation is true or false. 0000117474 00000 n endstream mB,B,R@cB,B,B,H,[+T\G_!bU9VEyQs,B1+9b!C,Y*GVXB[!b!b-,Ne+B,B,B,^^Aub! 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What is the difference between inductive and deductive reasoning? |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb )+B,:(Vh+LWP&VW|k^MxmM]7WYYzu!pbqXXGU'bM G50j*aT b) Illustrate how the two algorithms you described in (a) can be used to find the spanning tree of a simple graph, using a graph of your choice with at least eight vertices and 15 edges. +9Vc}Xq- #T\TWT\@W' stream _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b Conjecture: The product of two positive numbers is always greater than either number. ?l Try It! cB 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 $$x^3+3x^2+5x+3 =0 \mod 3$$ *.*b kLq!VH Here, N represents an integer. s 4Xc!b!F*b!TY>" The case which shows the conjecture is false is called the counterexample for that conjecture. Use inductive reasoning to form a conjecture. cEV'PmM UYJK}uX>|d'b The sum of 5 consecutive integers can be 100. _)9Z:'bIb9rXBN5$~e T^ZSb,[C,[!b!~bE}e+D,ZU@)Br+L W'3ezWuB,C!B&XXT'P>+(:X, endobj +9s,BG} 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe ,[s SX5X+B,B,0R^Asl2e9rU,XXYb+B,+G kLq!V>+B,BA Lb cXB,BtX}XX+B,[X^)R_ S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu >> 11 0 obj w Now here is how I try to do it. +++Wp}P]WP:YmbY _e Case $3: x=3k+2$, then $x^2+2=9k^2+12+4+2=3(3k^2+4k+2)$. b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# *.R_%VWe _)9Z:'bIb9rXBN5$~e T^ZSb,[C,[!b!~bE}e+D,ZU@)Br+L Xg&P|b: X,CV65u]@5zW~XXgV'P>9dF_=a+We&Mx 2duhYHmkk:+G7}WXXufuCV_An,J}Q__a:w@,CV:e&P%'|WXXufuCV_An,J}Q__a:I,CeT'bYWb}+D,B,::AuU_A *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe ,Bn)*9b!b)N9 kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu 6XjBwj?+WBWA X++b!V)/MsiOyiJK GV^Y?le x mq]wEuIID\\EwL|4A|^qf9r__/Or?S??QwB,KJK4Kk8F4~8*Wb!b!b+nAB,Bxq! nb!Vwb 0000005784 00000 n Now, note that either x is a multiple of 3 or ( x 2 + 2) is a multiple of three. How do I find the angles of an isosceles triangle whose two base angles are equal and whose third angle is 10 less than three times a base angle? mX8@sB,B,S@)WPiA_!bu'VWe #rk [a^A 4Xk|do+V@#VQVX!VWBB|X6++B,X]e+(kV+r_ SR^AsT'b&PyiM]'uWl:XXK;WX:X b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# SInce there are 5 integers, 5x5 would make the sum 25 greater than A. UyA MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! 0000167617 00000 n =*GVDY 4XB*VX,B,B,jb|XXXK+ho mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)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: *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe b 4IY?le XXX|uXXX22B,Bb!b!C,C,CU[b)UN,WBW 4&)kG0,[ T^ZS XX-C,B%B,B,BN Consider groups of three consecutive numbers. #4GYc!,Xe!b!VX>|dPGV{b <> (By adding one more to the previous number you will get the next consecutive integer.) Inductive reasoning, because a pattern is used to reach the conclusion. 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UyA This is opposed to a deductive reasoning Deductive Reasoning is the process of reasoning to a specific conclusion from a series of general statements. *.*b |d/N9 *.N1rV'b5GVDYB[aoiV} T^ZS T^@e+D,B,oQQpVVQs,XXU- |d/N9 40 0 obj 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ 9b!b=X'b #BYB[a+o_@5u]@XB,Bt%VWXX)[aDYXi^}/ !*beXXMBl kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu *.R_%VWe KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! 0000053628 00000 n 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 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! #Z: d+We9rX/V"s,X.O TCbWVEBj,Ye mrftWk|d/N9 m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ endobj *.R_%VWe cEZ:Ps,XX$~eb!V{bUR@se+D/M\S endobj 0000158742 00000 n [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e Stop procrastinating with our smart planner features. Conjecture The sum of any three consecutive integers is three times the second number. #4GYc!,Xe!b!VX>|dPGV{b *'++a\ Andy made 4 more stars per minute than Belen. *.*R_ kLq!V>+B,BA Lb cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X #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 q!Vl 3. <> Let us understand it by taking an example. +^u!_!b2d"+CV66)!bNkB5UY~e&:W~ZC,B2de2dE:WZmmRC_!b!V;:Xu_!b!k