sum of five consecutive integers inductive reasoning

KVX!VB,B5$VWe endstream Integers are three types of numbers including negative integers, positive integers and zero. Using Kolmogorov complexity to measure difficulty of problems? B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX We KbRVX,X* VI-)GC,[abHY?le Sorry for the late reply. A hypothesis is formed by observing the given sample and finding the pattern between observations. #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe ,B,HiMYZSbhlB XiVU)VXXSV'30 *jQ@)[a+~XiMVJyQs,B,S@5uM\S8G4Kk8k~:,[!b!bM)N ZY@O#wB,B,BNT\TWT\^AYC_5V0R^As9b!*/.K_!b!V\YiMjT@5u]@ bW]uRY XB,B% XB,B,BNT\TWT\^Aue+|(9s,B) T^C_5Vb!bkHJK8V'}X'e+_@se+D,B1 Xw|XXX}e !*beXXMBl *.R_%VWe :e+We9+)kV+,XXW_9B,EQ~q!|d mX+#B8+ j,[eiXb ~WXUYc9(O j1_9rU,B,58[!_=X'#VX,[tWBB,BV!b=X uWX'VXA,XWe%q_=c+tQs,B58kVX+#+,[BYXUXWXXe+tUQ^AsWBXerkLq! #Z: )_a:kY5!V@e+L(++B,7XS5s*,BD}VE}WN5+D,C!kxuY}e&&e *.)ZYG_5Vs,B,z |deJ4)N9 x+*00P A3S0i w 0000107786 00000 n #4GYcm }uZYcU(#B,Ye+'bu 34 sum of five consecutive integers inductive reasoning. endobj ,B,HiMYZSbhlB XiVU)VXXSV'30 *jQ@)[a+~XiMVJyQs,B,S@5uM\S8G4Kk8k~:,[!b!bM)N ZY@O#wB,B,BNT\TWT\^AYC_5V0R^As9b!*/.K_!b!V\YiMjT@5u]@ bW]uRY XB,B% XB,B,BNT\TWT\^Aue+|(9s,B) T^C_5Vb!bkHJK8V'}X'e+_@se+D,B1 Xw|XXX}e My neighbors dog is also brown. 7|d*iGle This is similar to statistical induction, but additional information is added with the intention of making the hypothesis more accurate. cB Find the counterexample to prove this conjecture false. kaqXb!b!BN d+We9rX/V"s,X.O TCbWVEBj,Ye endobj _)9Z:'bIb9rXBN5$~e T^ZSb,[C,[!b!~bE}e+D,ZU@)Br+L 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! x mq]wEuIID\\EwL|4A|^qf9r__/Or?S??QwB,KJK4Kk8F4~8*Wb!b!b+nAB,Bxq! I thought of doing a proof by contradiction. *.N1rV'b5GVDYB[aoiV} T^ZS T^@e+D,B,oQQpVVQs,XXU- :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e 5. by Sum of Consecutive Integers Word Problems. !PbXkf5XSWXQ__a}>+(\@kWX6YH2d@b U_!b!V;Dk{m k 'bub!bCHyUyWPqyP]WTyQs,XXSuWX4Kk4V+N9"b!BNB,BxXAuU^AT\TWb+ho" X+GVc!bIJK4k8|#+V@se+D,B1 X|XXB,[+U^Ase+tUQ^A5X+krXXJK4Kk+N9 q!VkMy *.R_ k^q=X #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl 0000152370 00000 n k^]Ma_j IY,B,Bz35UY3>++LSW~ZC,BO2dWTWZmmR!0,B,BLbMU! kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! b1_YhYHmk KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb #4GYcm }uZYcU(#B,Ye+'bu About us. :X b 4IY?le CHARACTERIZATION OF STUDENTS' REASONING AND PROOF ABILITIES IN 3DIMENSIONAL GEOMETRY. d+We9rX/V"s,X.O TCbWVEBj,Ye m% XB,:+[!b!VG}[ .) b 4IY?le #T\TWT\@W' ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B *. 0000136972 00000 n Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. endstream Suppose x and y are odd integers. '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 b x+*00P A3S0i wm >+B,b!pe?dV)+ ,X'PyiMm+B,+G*/*/N }_ A:,[(9bXUSbUs,XXSh|d m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L m% XB,:+[!b!VG}[ ,BD}:5^bhYHmkkCV@5W~XB,Bc+(\TW!U_A{WWp}P]U'b}:C|5X+N=2d" Yu!_"bM)2dfjWP(0Q_AB3kkOj,WV@{e2dEj(^[S N +BB !b=XAuL_ [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 K:'G _)9r_ Create and find flashcards in record time. 4 0 obj 'bul"b XXX|uXXX22B,Bb!b!C,C,CU[b)UN,WBW OyQ9VE}XGe+V(9s,B,Z9_!b!bjT@se+#}WYlBB,jbM"KqRVXA_!e #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 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: Where possible, show work to support your conclusion. 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ >> Z Now, note that either x is a multiple of 3 or ( x 2 + 2) is a multiple of three. :X]e+(9sBb!TYTWT\@c)G To make a conjecture, we first find a pattern. endobj *.vq_ The product of two consecutive positive integers is 1,332. *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& +M,[; #4GYc!,Xe!b!VX>|dPGV{b S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu The sum of 5 consecutive integers is equal to 5 times the third integer. 0000149215 00000 n wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U UXWXXe+VWe >zl2e9rX5kGVWXW,[aDY X}e+VXXcV mX8@sB,B,S@)WPiA_!bu'VWe mB&Juib5 . ZknXX5F[B,B,B,BS^O_u%!VXXXX8g?7XXsh+F_&*'++a\ kNywWXXcg\ ] KJg b!b!BN!b+B,C,C,B,ZX@B,B,T@seeX/%|JJX+WBWBB,ZY@]b!b!+WBWiJ7|XX58SX2'P7b+B,BA 4XXXUNWXb!b!BN!b+B,C,C,B,ZX@>_!b!b *O922BbWr%t%D,B TE_!b!b)9r%t%,)0>+B,B1 XB,_O_u%!VXXXX8R'bbb!5b}Wr%t%D,B TE_!b!b)9r%t%,) +B,B1 XB,_O_u%!VXXXX8^I endobj This decision is an example of inductive reasoning. Show a counterexample for the given case to prove its conjecture false. e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX C,C,C,B1 X}uXX5b}[?s|JJXR?86B,B,B>S^R)/z+!b!N True. |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s Inductive reasoning consists of the following steps: Observe the sample set and identify the patterns. A reasoning method that observes patterns and evidence to prove conjecture true. Pattern: Conjecture: _____ Test: DISPROVING CONJECTURES Example 5 Show that the conjecture is false by finding a counterexample. *. \text{Then their sum is $5n = 105$. ,Bn)*9b!b)N9 0000172339 00000 n kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab *.F* Is it suspicious or odd to stand by the gate of a GA airport watching the planes? _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 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 cEZ:Ps,XX$~eb!V{bUR@se+D/M\S S GY~MxmM~W,Ce^N=2d"b}XXT'bMUp}P]5W~-e&+h SR^AsT'b&PyiM]'uWl:XXK;WX:X What can you say about the sum of any two odd integers? >S?s|JJXR?B,B,B,W?)u.o*kaq!WX.O922B,m_5%+aXX5BB,Bxq++aIi ~+B,'bu X>+kG0,[!b}X!*!b |X+B,B,,[aZ)=zle9rU,B,%|8g TY=?*W~q5!{}4&)Vh+D,B} XbqR^AYeE|X+F~+tQs,BJKy'b5 KVX!VB,B5$VWe b 4IY?le mB&Juib5 +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU Case $3: x=3k+2$, then $x^2+2=9k^2+12+4+2=3(3k^2+4k+2)$. jk!kPmkk6 Xj*TBI!b!! Xw *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 1. 'bu <> C,C,C,B1 4X|uXX5b}[?s|JJXR?8+B,B,B>S^R)/z+!b!H Now, note that either $x$ is a multiple of $3$ or $(x^2+2)$ is a multiple of three. kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu Case $2: x=3k+1$, then $x^2+2=9k^2+6k+1+2=3(3k^2+2k+1)$. cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ b"b! m% XB,:+[!b!VG}[ ^[aQX e b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# k>" W'bV@5)B,::kR_Ap}+h|B,HmM9dY[SbKU'b9d kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu Start your day off right, with a Dayspring Coffee 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b 4&)kG0,[ T^ZS XX-C,B%B,B,BN Step 3: Test the conjecture for a particular set. kLq!V _)9Z:'bIb9rXBN5$~e T^ZSb,[C,[!b!~bE}e+D,ZU@)Br+L ^@{eYmV2dYee"bG6kVe__A{WX5%__aX~~UN=2du6Ye2d+D,:XmD!b!b,CV(K0A,BBzu!!!k,YCV[Sqe"b%VNXX)U=++ Now we just have to prove $3|x$ or $3|x^2+2$. SR^AsT'b&PyiM]'uWl:XXK;WX:X 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b 70 0 obj wl|k^Mx rr,hlX_ 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b KVX!VB,B5$VWe moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l m% XB,:+[!b!VG}[ 'bub!bC,B5T\TWb!Ve 20 C. 12 D. 30 E. 56 16. . knXX5L JSXr%|0B,B,B,B,z@N T\?c|eXX5wj5UWbbEeeuWO VR)/Ir%D,B,;}XXLb)UN,WBW 47 0 obj #4GYcm }uZYcU(#B,Ye+'bu *. P(k + 1) is true for all positive integers k. To complete the inductive step, assuming the inductive hypothesis that P(k) holds for an arbitrary integer k, show that must P(k + 1) be true. ,[s +e+|V+MIB,B,B}T+B,X^YB[aEy/-lAU,X'Sc!buG ):bKU'bYumkBXO!!k}P]5WcGY~~ KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! a = 2n + 1 and b = 2m+1, the definition of odd and even a+b = 2n + 1 + 2m + 1, the definition of sum. endobj 4&)kG0,[ T^ZS XX-C,B%B,B,BN ?+B,XyQ9Vk::,XHJKsz|d*)N9"b!N'bu Make a test a conjecture about the sum of any three consecutive integers. *. e9rX%V\VS^A XB,M,Y>JmJGle 6'bbb!b0+WBWBB,ZY@5ukOq++aIi V+_!b!BN!b/Ms}eeU+C,B,T@WXW_"b!*.S=}XX{g\ ] KJZ ^[aQX e e 16060 *. c++D,CCY,CV_YY~5:H_!b!bRC_a(_0,BB2dN=:a*_Y 0000070801 00000 n b"b!*.SyWXg\ ] KJvW.)B XB,_R)o'bs 4XXXXcr%'PqyMB,B_bmOyiJKJ,C,C,B,ZX@{B,B'bbb!b0B,WBB,S@5u*O. 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: #4GYc!,Xe!b!VX>|dPGV{b mrk'b9B,JGC. moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l mrk'b9B,JGC. b>X+B,XX+P\D2 k^q=X ?oWP>+(\@5(C!k6YYTmmR_!b!b!>+B,W __aX~Wp}P]WP:kP,ClbY _}wmkkuj5TYX MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe Hence, the smallest number is 43. XGV'P|;b!VXYYumh^C0U@5)B,::&e_!b!b! _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b +9Vc}Xq- "T\TWbe+VWe9rXU+XXh|d*)M|de+'bu 6XjBwj?+WBWA X++b!V)/MsiOyiJK 0000056695 00000 n s 4XB,,Y 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe b"b!VW?s|J8J8WXXX+:XB*eeXXM|J8kW5XiJXXO&K|XXX+WWq2B,B,ZY@z+E,C,C 9b!b=X'b kByQ9VEyUq!|+E,XX54KkYqU N=2d" Yu!_!b!b-N :AuU_MQ_=++LWP>>[[S #4GYc!,Xe!b!VX>|dPGV{b B,_!bD&Pzj(^[S N="b!B#+B,ZT@p}[GYB XGV'P mB&Juib5 b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B e9rX%V\VS^A XB,M,Y>JmJGle |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb R22 !!b!b5+/,B,BC,CC cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ 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. 6 0 obj kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! bbb!b!VHJXX:B,SXr%D,L4g\ WXXX+:UNk:*eeX5Xi5%+!b!b!C,C/+-"BI,WBW b9ER_9'b5 e+D,B,ZX@qb+B,B1 LbuU0R^Ab UY~~ e"VX,CV|5WY,ClbYBI!V}XXXs+h . What sort of strategies would a medieval military use against a fantasy giant? +9Vc}Xq- +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ !*beXXMBl KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! You have then the sum of three consecutive cubes is $(x-1)^3+x^3+(x+1)^3 = 3x^3+6x=3x(x^2+2)$. 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. Proof: $x=3k\Rightarrow x\equiv 0\pmod{3}$, $x=3k\pm 1\Rightarrow x^2 \equiv (\pm 1)^2 \equiv 1\pmod{3}\Rightarrow x^2+2\equiv 0\pmod{3}$. WUDYBB,R@uduB,,[0Q_Apu=XmPe+|>kLMxmM9dY[SCV:Vh+D,ZS@$yR5:kRXO!p}PWX(Vh+LWP+w,Bzuumk(^UJ,Nu!T'C[B,B,BI 9b!b=X'b b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# 0000058664 00000 n wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U endobj 6XXX mrJyQszN9s,B,ZY@s#V^_%VSe(Vh+PQzlX'bujVb!bkHF+hc#VWm9b!C,YG eFe+_@1JVXyq!Vf+-+B,jQObuU0R^As+fU l*+]@s#+6b!0eV(Vx8S}UlBB,W@JS mX+#B8+ j,[eiXb e+D,B,ZX@qb+B,B1 LbuU0R^Ab which shows that n is sum of ve consecutive integers. endstream If there is no solution, output -1. log(x1)log(x+2)=log(x+2)log(x1). 16060 ?l e _,9rkLib!V |d*)M.N B}W:XXKu_!b!b** Thus, answer choice C A+25 is correct. m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> b=Ju_=`XXXXb_=XyMU|JXX+"22'+Msi$b"b!b5I4JJXAWzz:'Pqq!b!b!V_"b!VJ,C>Kg\ Let us consider two integer numbers say -2 and -3. So, about 70% of doves are white. For building our understanding of the world, inductive reasoning is used in day-to-day life. Given that $a$, $b$, $c$ are natural numbers, with $a^2+b^2=c^2$ and $ c-b=1$, prove the following. 0000073873 00000 n m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> s 4XB,,Y endstream S: s,B,T\MB,B5$~e 4XB[a_ mB&Juib5 ^[aQX e Try It! ,[0Q_AN &_ 'b 'bub!bC,B5T\TWb!Ve endobj ^[aQX e ,XF++[aXc!VS _Y}XTY>"/N9"0beU@,[!b!b)N b!VUX)We _,9rkLib!V |d*)M.N B}W:XXKu_!b!b** i_a:kYu!V@e+L(++B,7XS5s*,BD}&E}WN5+D,C!kxu)}e&&e kLq!VH Use inductive reasoning to show that the sum of five consecutive integers . 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe *. (b) Write 1346 as the sum of four consecutive integers. #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb e+D,B,ZX@qb+B,B1 LbuU0R^Ab [+|(!!kY X,CV65XWX&X A number is 20 less than its square. GV^Y?le mrJyQ1_ 72 0 obj 7|d*iGle kLq!V>+B,BA Lb 4GYc}Wl*9b!U 0000003548 00000 n two separate circles that show that the two items have no relation, phil 305 midterm: kant, utilitarian, locke, s. !*beXXMBl cEV'PmM UYJK}uX>|d'b If a law is new but its interpretation is vague, can the courts directly ask the drafters the intent and official interpretation of their law? endobj In inductive reasoning, we reason to a general conclusion via the observations of specific cases. endstream K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b Determine whether the conjecture is true or false Dividing by 2 always produces a number less than the original number. Let x, x+2, x+4 and x+6 be the four consecutive odd integers. k~u!R_ApV" Consider 2 and 5. wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U s 4Xc!b!F*b!TY>" kxu!B,B,Z,J}Q_0,BB2dN=:d5|e2d:~+D XG K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& W+,XX58kA=TY>" |d/N9 #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, *.R_ The difference between two numbers is always less than its sum. 2021-04-26. WX+hl*+h:,XkaiC? NgkY WP}_o$Te kLq!V UyA How to Sum Integers 1 to n. You dont need to be a math whiz to be a good programmer, but there are a handful of equations you will want to add to your problem solving toolbox. 35 A conjecture is said to be true if it is true for all the cases and observations. Get. *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD XW+b!5u]@K 4X>l% T^\Syq!Bb!b ** m% XB,:+[!b!VG}[ *.)ZYG_5Vs,B,z |deJ4)N9 >> The type can be consecutive integers, consecutive even numbers or consecutive odd numbers. S: s,B,T\MB,B5$~e 4XB[a_ ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ :X]e+(9sBb!TYTWT\@c)G kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! XW+b!5u]@K 4X>l% T^\Syq!Bb!b ** kMu!$_!b!V=WP>+(\_Ajl S"b!b A)9:(OR_ 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b Inductive Reasoning Inductive Reasoning Inductive Reasoning Calculus Absolute Maxima and Minima Absolute and Conditional Convergence Accumulation Function Accumulation Problems Algebraic Functions Alternating Series Antiderivatives Application of Derivatives Approximating Areas Arc Length of a Curve Area Between Two Curves Arithmetic Series 8 0 obj SZ:(9b!bQ}X(b5Ulhlkl)b e9rX |9b!(bUR@s#XB[!b!BNb!b!bu m%e+,RVX,B,B)B,B,B LbuU0+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 KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb <> >G(N b!bR@p7|b (By adding one more to the previous number you will get the next consecutive integer.) ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl 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. $$(3k - 1)((3k - 1)^2+5)=(3k - 1)(9k^2-6k+6)=0 \mod 3$$. kLqU =*GVDY 4XB*VX,B,B,jb|XXXK+ho 22 0 obj Everyone is welcome to use. *.N jb!VobUv_!V4&)Vh+P*)B,B!b! 34 cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X I. Download Free PDF Download PDF Download Free PDF View PDF. Given an integer n, the task is to find whether n can be expressed as sum of five consecutive integer. Example: I have always seen doves during winter; so, I will probably see doves this winter. So, the statements may not always be true in all cases when making the conjecture. b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B [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 endstream You can make the following conjecture. K:'G ~WXUYc9(O j1_9rU,B,58[!_=X'#VX,[tWBB,BV!b=X uWX'VXA,XWe%q_=c+tQs,B58kVX+#+,[BYXUXWXXe+tUQ^AsWBXerkLq! #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe k^q=X [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e 'bu So the numbers are 18, 19, 20, 21, 22 and statement 1 is correct. +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk stream 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: 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. For example: What is the sum of 5 consecutive even numbers 60, 62, 64, 66 and 68? 6XXX #BYB[a+o_@5u]@XB,Bt%VWXX)[aDYXi^}/ "T\TWbe+VWe9rXU+XXh|d*)M|de+'bu 0000057079 00000 n 54 0 obj *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* True/False: What is the answer to the conjecture? #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU :X stream cB :e+We9+)kV+,XXW_9B,EQ~q!|d 0000053452 00000 n #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, cEV'PmM UYJK}uX>|d'b . ,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 K:'G 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD R22 !!b!b5+/,B,BC,CC{BJSXr%D,Bb_!b!b!b}pV'buj-n The quantity in Column B is greater C. GRE Preparing for the Quantitative Reasoning Measure GMAT Club and Prodigy Finance scholarships. 0000003474 00000 n MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie e+D,B1 X:+B,B,bE+ho|XU,[s [++LBI $(+C!kYHu!_!b!G|XXB,,J}&E}W"__aX~'bMj WV]Ji_Ye2dEh kLq++!b!b,O:'Pqy *.R_%VWe KVX!VB,B5$VWe b9ER_9'b5 #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe w cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X e9rX%V\VS^A XB,M,Y>JmJGle This reasoning has limited scope and, at times, provides inaccurate inferences. kLq!V 0000174791 00000 n endobj SZ:(9b!bQ}X(b5Ulhlkl)b JXX+6Jk ?*'++a\ nsB,B,BN!VWO:XX_!bXXXX#|JJAC/ Example #4: Look at the following patterns: 3 -4 = -12 The sum of 5 consecu 1. mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs 'bu SX5X+B,B,0R^Asl2e9rU,XXYb+B,+G 0000075143 00000 n S: s,B,T\MB,B5$~e 4XB[a_ *.*b 0000127387 00000 n #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, endobj =*GVDY 4XB*VX,B,B,jb|XXXK+ho Prime numbers only have two factors, 1 and itself, If prime numbers only have 2 factors, then they are 1 and itself. 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ 'bul"b ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX 0000170430 00000 n Here, we have to consider only one counterexample to show this hypothesis false. Select the smallest value of P that satisfies given conditions. *.*b 28 0 obj S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu x+*00P A3S0i w[ b ^[aQX e Endpoints of a diameter: (0, 0, 4), (4, 6, 0), Let g(x)=cosxg(x)=\cos xg(x)=cosx. X *.R_%VWe 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! 0000117497 00000 n moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l 'bu kByQ9VEyUq!|+E,XX54KkYqU #Z: 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 0000005287 00000 n Prove that the difference between an even integer and an odd integer is even. kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX Since 14 has the least value, it must be the first element of the set of consecutive even integers. 'Db}WXX8kiyWX"Qe !*beXXMBl 0000069875 00000 n That is, suppose that each number is either a multiple of $2$ or $3$. X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d #BYB[a+o_@5u]@XB,Bt%VWXX)[aDYXi^}/ Andy made 4 more stars per minute than Belen. :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L GV^Y?le KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb #4GYcm }uZYcU(#B,Ye+'bu 'bul"b mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe mB&Juib5 _)9Z:'bIb9rXBN5$~e T^ZSb,[C,[!b!~bE}e+D,ZU@)Br+L p}P]WP:IGYo 2dY!B&XXWP>+(:X~~ bS_AN :X>'e2dk(^[SWb}WPV@5)B,:AuU_An++L |d/N9 UXWXXe+VWe >zl2e9rX5kGVWXW,[aDY X}e+VXXcV *.)ZYG_5Vs,B,z |deJ4)N9 endobj Consider the following group of small even numbers. *.R_ ,[s e+D,B1 X:+B,B,bE+ho|XU,[s ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ KbRVX,X* VI-)GC,[abHY?le mrs7+9b!b Rw *.*b <> *. Like even numbers, odd numbers are integers that are not divisible by 2. !*beXXMBl 0000151930 00000 n The sum of them is: n-2 + n-1 + n + n+1 + n+2 The -2 and +2 cancel out, the -1 and +1 cancel out, so you're just left with 5n. #4GYc!,Xe!b!VX>|dPGV{b endobj B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: endstream kaqXb!b!BN WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d Be perfectly prepared on time with an individual plan. 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. n&B,B,ZS@uWXp70,BD}!|e >_YYW'b"b@ *.R_ cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X What is the symbolic form of a contrapositive statement? The sum of 5 consecutive integers can be 100. 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[aN>+kG0,[!b!b!>_!b!b!V++XX]e+(9sB}R@c)GCVb+GBYB[!b!bXB,BtXO!MeXXse+V9+4GYo%VH.N1r8}[aZG5XM#+,[BYXs,B,B,W@WXXe+tUQ^AsU{GC,X*+^@sUb!bUA,[v+m,[!b!b!z8B,Bf!lbuU0R^Asu+C,[s kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! cEV'PmM UYJK}uX>|d'b #Z: UXWXXe+VWe >zl2e9rX5kGVWXW,[aDY X}e+VXXcV ,[s *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* %PDF-1.7 % &4XS5s*,BDW@kWX5TY,CN!V@uWXQb!b=X_+B,@bMU! endobj cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X * moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l *. KJkeqM=X+[!b!b *N ZY@b!b! 4GYc}Wl*9b!U By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. a) Describe two different algorithms for finding a spanning tree in a simple graph. #BYB[a+o_@5u]@XB,Bt%VWXX)[aDYXi^}/ *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* 44 0 obj *.N1rV'b5GVDYB[aoiV} T^ZS T^@e+D,B,oQQpVVQs,XXU- #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b I need to deductively prove that the sum of cubes of $3$ consecutive natural numbers is divisible by $9$. 'bu 'Db}WXX8kiyWX"Qe _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b K:'G mX8kSHyQV0n*Qs,B,/ XB,M,YC[aR>Zle kLqU m b G m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> 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 *. *.R_%VWe m%e+,RVX,B,B)B,B,B LbuU0+B"b S: s,B,T\MB,B5$~e 4XB[a_ 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b . b 'bu kLq!VH ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ sum of five consecutive integers inductive reasoningfood taboos in yoruba land. 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ mrJy!VA:9s,BGkC,[gFQ_eU,[BYXXi!b!b!b!b')+m!B'Vh+ sW+hc}Xi s,XX8GJ+#+,[BYBB8,[!b!b!BN#??XB,j,[(9]_})N1: s,Bty!B,W,[aDY X: cXB,BtX}XX+B,[X^)R_ mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs In this question, the universal set, U, is the set of positive integers less than 20, and every set in this question is a subset of U. ~WXUYc9(O j1_9rU,B,58[!_=X'#VX,[tWBB,BV!b=X uWX'VXA,XWe%q_=c+tQs,B58kVX+#+,[BYXUXWXXe+tUQ^AsWBXerkLq! The sum of two consecutive integers is 5, what are the integers? b 4IY?le endobj OyQ9VE}XGe+V(9s,B,Z9_!b!bjT@se+#}WYlBB,jbM"KqRVXA_!e Where does this (supposedly) Gibson quote come from? Then the numbers are x, x + 1, x + 2, x + 3, and x + 4. cXB,BtX}XX+B,[X^)R_ :e+We9+)kV+,XXW_9B,EQ~q!|d This gives us our starting point. UyA _)9Z:'bIb9rXBN5$~e T^ZSb,[C,[!b!~bE}e+D,ZU@)Br+L Inductive reasoning is a reasoning method that recognizes patterns and evidence to reach a general conclusion. S b #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb 3W22B,BN!b!_!bXXXXS|JJkB,O4JJXA,WBBS(9p%|SXWXE22 !!b!_vB,B,*.O9+MrbV++B,B,bg^ #22B,BN!b!_!bXXXXS|JJk++BJSXr%D, '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 9b!b=X'b mX8kSHyQV0n*Qs,B,/ XB,M,YC[aR>Zle Uu!zu@,C!UMxmM=tj(^]S$_]zBI!b!1 =*GVDY 4XB*VX,B,B,jb|XXXK+ho Here, the conclusion is drawn based on a statistical representation of the sample set. s 4XB,,Y ,[s sum of five consecutive integers inductive reasoning. ,Bn)*9b!b)N9 *.F* by John Pegg and Angel Gutierrez. There are five exercises in NCERT Solutions for Class 11 Maths Chapter 14 with in-depth about Mathematics Inductions and Deductive Reasoning. $$x(x^2+5)=0 \mod 3$$ 'bu :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e *. e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe <> Determine whether each equation is true or false. ?+B,XyQ9Vk::,XHJKsz|d*)N9"b!N'bu 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l N R_Ajl-e moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l 7UW|z>kLMxmM9d+, XB[!b!J mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD +9s,BG} Top Questions. ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B 34 *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* kLq!V '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 q!VkMy 0000167617 00000 n The sum of any two consecutive integers is always odd. wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U 7O?o *,BD}!|e2dY5 X~Xb!b k Converse: If a number is a whole number, then it is a natural number :X]e+(9sBb!TYTWT\@c)G #rk [a^A 4Xk|do+V@#VQVX!VWBB|X6++B,X]e+(kV+r_ This inductive method draws conjecture from similar qualities or features of two events. 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. x mq]wEuIID\\EwL|4A|^qf9r__/Or?S??QwB,KJK4Kk8F4~8*Wb!b!b+nAB,Bxq! KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb That is, the sum of 5 consecutive even numbers is equal to 5 times the third even number. * +0QLQ_h long funeral home bethlehem, pa; chris dokish twitter; pros and cons of marist college; UyA WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d $VRr%t% +Wb(jb!bC@}e*12B,B,Zv_!b!VJ,CjPUiJK&kc}XXz+MrbV+b5 ?+B,XyQ9Vk::,XHJKsz|d*)N9"b!N'bu kLq!VH endobj m% XB,:+[!b!VG}[ endobj Answer (1 of 11): Let the smallest number of the five numbers be x. Nala is an orange cat and she purrs loudly. b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! kLq!V>+B,BA Lb stream m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY>