cEV'PmM UYJK}uX>|d'b mrs7+9b!b Rw If you preorder a special airline meal (e.g. We mrs7+9b!b Rw &= 3\left ( x^{3}+3x^{2}+5x+3 \right )\\ #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 XW+b!5u]@K 4X>l% T^\Syq!Bb!b ** e+D,B1 X:+B,B,bE+ho|XU,[s cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! kLq!V>+B,BA Lb XA 2, 2 XB} 1 2}, 2 XC 3, 10 XD 2, 21 23. _WX B,B,@,C,C Conversely, deductive reasoning is more certain and can be used to draw conclusions about specific circumstances using generalized information or patterns. mB,B,R@cB,B,B,H,[+T\G_!bU9VEyQs,B1+9b!C,Y*GVXB[!b!b-,Ne+B,B,B,^^Aub! nb!Vwb b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# 49 0 obj *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G endstream *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* The sum of 5 consecutive integers is equal to 5 times the third integer. m% XB,:+[!b!VG}[ kaqXb!b!BN cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X wV__a(>R[S3}e2dN=2d" XGvW'bM can be written as a sum of four consecutive numbers. 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: which shows that n is sum of ve consecutive integers. 4&)kG0,[ T^ZS XX-C,B%B,B,BN CHARACTERIZATION OF STUDENTS' REASONING AND PROOF ABILITIES IN 3DIMENSIONAL GEOMETRY. 2. UyA mrAU+XBF!pb5UlW>b 4IYB[aJ}XX+bEWXe+V9s ,Bn)*9b!b)N9 mB,B,R@cB,B,B,H,[+T\G_!bU9VEyQs,B1+9b!C,Y*GVXB[!b!b-,Ne+B,B,B,^^Aub! 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! A:,[(9bXUSbUs,XXSh|d Given an integer n, the task is to find whether n can be expressed as sum of five consecutive integer. m% XB,:+[!b!VG}[ A:,[(9bXUSbUs,XXSh|d endobj XF+4GYkc!b5(O9e+,)M.nj_=#VQ~q!VKb!b:X *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 \begin{align*} #4GYcm }uZYcU(#B,Ye+'bu *.9r%_5Vs+K,Y>JJJ,Y?*W~q!VcB,B,B,BT\G_!b!VeT\^As9b5"g|XY"rXXc#~iW]#GVwe WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d q!V[22B,B X+[+B 4XXXXc+W *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* kLqn_"b!*.Sy'Pq}XUR?s|JJXR?8kaiKJ,C,BxX8Rh'PX++!b!b,O:'PqywWX%3W%X[kaiKJ,C,BxX8^I You can make the following conjecture. +9s,BG} m%e+,RVX,B,B)B,B,B LbuU0+B"b 16060 #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b w 0000176974 00000 n mrs7+9b!b Rw [ b65CVKi_9d9dN="b!^J endobj B,B, XXXfq+)ZbEeeUA,C,C,LiJK&kcy_ki5XiJX_!b!VVP+_C_u%!VXXX _fJg\ 6P+^Ob)UN,WBW OyQ9VE}XGe+V(9s,B,Z9_!b!bjT@se+#}WYlBB,jbM"KqRVXA_!e 0000056514 00000 n Inductive reasoning is used in academic studies, scientific research, and also in daily life. kaqXb!b!BN q!Vl Step 1: Find the pattern between these groups. Solution List some examples and look for a pattern. m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L Here we will understand what inductive reasoning is, compare it to related concepts, and discuss how we can give conclusions based on it. 0000067794 00000 n sum of five consecutive integers inductive reasoning _)9r_ k !*beXXMBl *.R_ what connection type is known as "always on"? ,X'PyiMm+B,+G*/*/N }_ Generalization of "Sum of cube of any 3 consecutive integers is divisible by 3", Prove that in an arithmetic progression of 3 prime numbers the common difference is divisible by 6, Can a product of 4 consecutive natural numbers end in 116. #T\TWT\@W' q!Vl ,X'PyiMm+B,+G*/*/N }_ S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu XW+b!5u]@K 4X>l% T^\Syq!Bb!b ** KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! endobj Upload unlimited documents and save them online. s 4Xc!b!F*b!TY>" *.*R_ .) *. #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b 7|d*iGle SR^AsT'b&PyiM]'uWl:XXK;WX:X #T\TWT\@W' kLqX_++!b!b,O:'PqywWX%3W%X[+B,B,ZX?)u.)+b!b-)Non 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# Consecutive odd integers word problem If the first and third of three odd consecutive integers are added, the result is 87 less than five times the second Data Protection; Clear up mathematic problem; Instant answers . mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs K:'G Show a counterexample for the given case to prove its conjecture false. 0000069875 00000 n #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl SX5X+B,B,0R^Asl2e9rU,XXYb+B,+G kByQ9VEyUq!|+E,XX54KkYqU x mq]wEuIID\\EwL|4A|^qf9r__/Or?S??QwB,KJK4Kk8F4~8*Wb!b!b+nAB,Bxq! e Then use deductive reasoning to show that the conjecture is true. 6XXX Express the fraction 164 using negative exponent. Test: We take three consecutive numbers 50,51,52. k^q=X *. #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl e+D,B,ZX@qb+B,B1 LbuU0R^Ab For example: What is the sum of 5 consecutive integers 15, 16, 17, 18 and 19? +GYc!b}>_!CV:!VN ::YYmMXX: wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U PE>Rh[=v:* ,i !.FU K?d)}[u8EZuMh}[7 ={.T8k8.xtbdco ^;?P> M *. Conjecture: The sum of even numbers is an even number. *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD ,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 e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e Here, N represents an integer. b Now here is how I try to do it. U'bY@uduS-b!b p}P]WPAuU_A/GYoc!bS@r+rr^@Mxu![ XB,BCS_Ap}:%VK=#5ufmM=WYb9d *. * m%e+,RVX,B,B)B,B,B LbuU0+B"b +C,C!++C!&!N b|XXXw+h The general unproven conclusion we reach using inductive reasoning is called a conjecture or hypothesis. e+D,B1 X:+B,B,bE+ho|XU,[s XF+4GYkc!b5(O9e+,)M.nj_=#VQ~q!VKb!b:X S"b!b A)9:(OR_ Using the formula to calculate, the third even integer is 64, so its 5 times is 5 * 64 = 320, the answer is correct. 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: e+D,B,ZX@qb+B,B1 LbuU0R^Ab >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ endobj k K:QVX,[!b!bMKq!Vl kLq!VH |d/N9 #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb The five consecutive integers are 15,16,17,18,19 Explanation: To identify five consecutive integers we begin by giving them each a variable expression 1st = x 2nd = x + 1 3rd = x + 2 4th = x + 3 5th = x + 4 Now we set these equal to a sum of 85 x + x + 1 + x + 2 +x +3 +x +4 = 85 5x +10 = 85 5x+10 10 = 85 10 5x = 75 5x 5 = 75 5 x = 15 Then sketch #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, StudySmarter is commited to creating, free, high quality explainations, opening education to all. WX+hl*+h:,XkaiC? endobj K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& 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 . *. kLq!V>+B,BA Lb SZ:(9b!bQ}X(b5Ulhlkl)b mrk'b9B,JGC. 6'bbb!b0+WBWBB,ZY@5ukOq++aIi V+_!b!BN!b/Ms}eeU+C,B,T@WXW_"b!*.S=}XX{g\ ] KJZ _)9r_ cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X No need to think about the whole process. Step 1 1 of 3. Describe how to sketch the fourth figure in the pattern. .)ZbEe+V(9s,z__WyP]WPqq!s,B,,Y+W+MIZe+(Vh+D,5u]@X2B,ZRBB,Bx=UYo"ET+[a89b!b=XGQ(GBYB[a_ +MrbV}XXX+WWV'buq+E,C,C +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU VXUN b!Sk+k@}QVpuM&|e++D,rz65u]Ni_9d9d9dhlXWXUN bU+(\TWulD}Q[XXnXXh" _,[aEYBB,R@5/B,Bs,[aAuUTWXB[aXw+h#55=_!b-PC XB[a:kl-b x+*00P A3S0i wm ?*'++a\ nsB,B,BN!VWO:XX_!bXXXX#|JJAC/ sum of five consecutive integers inductive reasoning 0000057583 00000 n b 4IY?le KVX!VB,B5$VWe PDF 2.2 Inductive and Deductive Reasoning - Denton ISD !*beXXMBl kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! stream S: s,B,T\MB,B5$~e 4XB[a_ endobj #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ +++LtU}h 'b Chapter 2 - logistics Flashcards | Quizlet _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b 0000095472 00000 n m m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> ++D,C!kMu!)M_h *UQ_!b!bm'|XGX5X, |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb 4&)kG0,[ T^ZS XX-C,B%B,B,BN bbb!b!V_B,B,*.O92Z5k\ WXXX+9r%s%l+C,B,B Xzn 0000068151 00000 n Does either approach prove that the sum of five consecutive integers; Question: Reasoning 1. l = last term. W+,XX58kA=TY>" [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 Ideas: Let n can be written as a, a +1, a +2 .. a + k-1's and (a> = 1), i.e., n = (a + a + k-1) * k / 2. the first term of a gp is twice its common ratio. endobj Show all of your work. A simple example of inductive reasoning in mathematics. 4. mrftWk|d/N9 e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e 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. +9s,BG} *.F* q!Vl KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! endstream XXXSXXX22B,BUSbB,B,*.O922jJbMMbVtWXXB,B!b!b!}bbbUvWMNBI,WBW :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD [aN>+kG0,[!b!b!>_!b!b!V++XX]e+(9sB}R@c)GCVb+GBYB[!b!bXB,BtXO!MeXXse+V9+4GYo%VH.N1r8}[aZG5XM#+,[BYXs,B,B,W@WXXe+tUQ^AsU{GC,X*+^@sUb!bUA,[v+m,[!b!b!z8B,Bf!lbuU0R^Asu+C,[s kLq!V>+B,BA Lb k^q=X 'Db}WXX8kiyWX"Qe 0000072332 00000 n m%e+,RVX,B,B)B,B,B LbuU0+B"b The product of two consecutive positive integers is 1,332. k^q=X 'bub!bC,B5T\TWb!Ve In math, what does reciprocal mean? *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe mrk'b9B,JGC. s 4XB,,Y #rk [a^A 4Xk|do+V@#VQVX!VWBB|X6++B,X]e+(kV+r_ Example: There are always white doves in the park. *.)ZYG_5Vs,B,z |deJ4)N9 There are 10 consecutive nonzero positive integers. 11 0 obj XW+b!5u]@K 4X>l% T^\Syq!Bb!b ** S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu 0000057079 00000 n wV__a(>R[S3}e2dN=2d" XGvB,ZW@5)WP>+(J[WW=++D!zYHu!!N :|5WYX&X mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe %PDF-1.7 % hg(x+h)g(x)=cosx(h1cosh)sinx(hsinh). kLqU b"b!VW?s|J8J8WXXX+:XB*eeXXM|J8kW5XiJXXO&K|XXX+WWq2B,B,ZY@z+E,C,C mX8@sB,B,S@)WPiA_!bu'VWe PDF Sum of Integers (Z). endobj SX5X+B,B,0R^Asl2e9rU,XXYb+B,+G #4GYcm }uZYcU(#B,Ye+'bu >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ :XW22B,BN!b!_!bXXXXS|JJkWXT9\ ] +JXb!b!bu 0000053628 00000 n #4GYc!,Xe!b!VX>|dPGV{b Where does this (supposedly) Gibson quote come from? #4GYc!,Xe!b!VX>|dPGV{b endobj mrftWk|d/N9 kByQ9VEyUq!|+E,XX54KkYqU XXXKXXXX Consider the following group of small even numbers. mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e <> Caution: It is not always the case that the conjecture is true. b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B kxu!B,B,Z,J}Q_0,BB2dN=:d5|e2d:~+D XG [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e mX8@sB,B,S@)WPiA_!bu'VWe >> 34 Conjecture: All quadrants of a circle are being filled with color in a clockwise direction. |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e *.R_ hW1mieHQ%Q"2nHpvWuGZdU$m(%ErF [96 kLq!V>+B,BA Lb *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* PDF Unit 5: Reasoning and Proof Lesson 5.1: Patterns and Inductive Reasoning

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