q!VkMy a) Describe two different algorithms for finding a spanning tree in a simple graph. [+|(>R[S3}e2dN=2d" XGvW'bM >X@{MxmM]W'|bWse+(VXX[V_!b!b!Te Stop procrastinating with our smart planner features. *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* .)ZbEe+V(9s,z__WyP]WPqq!s,B,,Y+W+MIZe+(Vh+D,5u]@X2B,ZRBB,Bx=UYo"ET+[a89b!b=XGQ(GBYB[a_ *.9r%_5Vs+K,Y>JJJ,Y?*W~q!VcB,B,B,BT\G_!b!VeT\^As9b5"g|XY"rXXc#~iW]#GVwe 0000186817 00000 n p}b A:,[(9bXUSbUs,XXSh|d Formula for sum of 'n' terms of an arithmetic sequence: S n = n 2 [ 2 a 1 + ( n - 1) d]. 4GYc}Wl*9b!U 22 0 obj XXX22B,E}JJB,O4JJXA,WBBjb}WXX) ,Bn)*9b!b)N9 ?l ?l bWjXXU\@_!k6*'++a\ szkEXXXo3}e5?C,B,B,BnB!VXXX22B*bWjXXU\@qbW"M4JJXA,WBz?"B!b!b!bY?! H\$56Nkxd}AnT?6P]H1DMa #" b"b!:*b!b53W%uT+B,jb!b!b =+C,C Example #4: Look at the following patterns: 3 -4 = -12 To To prove that a conjecture is true, you need to prove it is true in all cases. mrs7+9b!b Rw 34 stream *. 4&)kG0,[ T^ZS XX-C,B%B,B,BN q!VkMy 18 0 obj 19 0 obj # 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 #T\TWT\@W' #4GYcm }uZYcU(#B,Ye+'bu #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe 'bu 0000151454 00000 n kLq!VH Answer (1 of 4): let x-2,x-1,x,x+1,x+2 are 5 consecutive integers sum is -5 soo =>x-2+x-1+x+x+1+x+1 =-5 =>5x=-5 => x=-1 x-2 = -3 x-1 = -2 x+1 = 0 x+2 = 1 therefore numbers are In this tutorial, you learned how to sum a series of consecutive integers with a simple and easy to remember equation. 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. ~iJWXX2B,BA Xm|XXhJ}J++!b!b,O:WXkOq!V22!b!b *N j+B,T@seeXU+W\ ] keyB,B=3W%X|XX{:Xu4!!VkPq!V_!b!C,C,C,BR_F|JJXX+Nb!b)9r%t%,)j+B,S@)B)un*|eXX mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab #4GYc!,Xe!b!VX>|dPGV{b >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ *.N1rV'b5GVDYB[aoiV} T^ZS T^@e+D,B,oQQpVVQs,XXU- 3W22B,BN!b!_!bXXXXS|JJkB,O4JJXA,WBBS(9p%|SXWXE22 !!b!_vB,B,*.O9+MrbV++B,B,bg^ #22B,BN!b!_!bXXXXS|JJk++BJSXr%D, cEV'PmM UYJK}uX>|d'b endobj B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX d+We9rX/V"s,X.O TCbWVEBj,Ye Sum of Multisets: The sum of two Then at least five computers are used by three or more students. x+*00P A3S0i wR e !*beXXMBl Is a PhD visitor considered as a visiting scholar? e9rX |9b!(bUR@s#XB[!b!BNb!b!bu *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe 6++[!b!VGlA_!b!Vl 10 0 obj e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX Try It! x mq]wEuIID\\EwL|4A|^qf9r__/Or?S??QwB,KJK4Kk8F4~8*Wb!b!b+nAB,Bxq! B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX Be perfectly prepared on time with an individual plan. :X 6XXX _WX B,B,@,C,C #BYB[a+o_@5u]@XB,Bt%VWXX)[aDYXi^}/ 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe +e+|V+MIB,B,B}T+B,X^YB[aEy/-lAU,X'Sc!buG s 4XB,,Y |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb *.N1rV'b5GVDYB[aoiV} T^ZS T^@e+D,B,oQQpVVQs,XXU- +9s,BG} 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 |d/N9 wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U e+D,B,ZX@qb+B,B1 LbuU0R^Ab 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: The product of two consecutive positive integers is 1,332. ~iJ[WXX2B,BA X;_!b!VijJ,W\ kNy}XXBN!b/MsqUWXX58knb!bh*_5%+aXX5HB,Bxq++aIi ~+^@)B)u.nj_bbU'bB,Bty!!!b!}Xb"b!*.Sy, 'b For example, the sum of 3 consecutive odd integers is 30, find these odd integers. !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!VWe [S@5&&PCCC,[kM ?+B,XyQ9Vk::,XHJKsz|d*)N9"b!N'bu kMuRC_a+B Although it looks a bit similar, there are still differences. cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ ?+B,XyQ9Vk::,XHJKsz|d*)N9"b!N'bu 34 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b Conjecture: The product of two positive numbers is always greater than either number. Let S be the number of perfect squares among the integers from 1 to 20136. *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe 70 0 obj ,Bn)*9b!b)N9 This. mB&Juib5 bbb!b!V_B,B,*.O92j=zk\ F endobj 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ K:'G :X]e+(9sBb!TYTWT\@c)G e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e ,[s k mrs7+9b!b Rw kLq!V>+B,BA Lb _b!b!V^XXU\@seeuWJXD,WBW +++LtU}h 6XXX 0000053628 00000 n ^[aQX e kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! For $$x=\pm 1 \mod 3$$, *. *.*b <> UyA #T\TWT\@W' #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 *.N jb!VobUv_!V4&)Vh+P*)B,B!b! |d/N9 +e+|V+MIB,B,B}T+B,X^YB[aEy/-lAU,X'Sc!buG *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD )_a:kY5!V@e+L(++B,7XS5s*,BD}VE}WN5+D,C!kxuY}e&&e Start your day off right, with a Dayspring Coffee ,X'PyiMm+B,+G*/*/N }_ 9 0 obj kByQ9VEyUq!|+E,XX54KkYqU kLq!VH Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. Case $3: x=3k+2$, then $x^2+2=9k^2+12+4+2=3(3k^2+4k+2)$. p}P]WP:IGYo 2dY!B&XXWP>+(:X~~ bS_AN :X>'e2dk(^[SWb}WPV@5)B,:AuU_An++L kLqU Here, the product of both the numbers is 10, which is positive. WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d e b 4IY?le What is the sum of the first 20 Z? mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab mX+#B8+ j,[eiXb nb!Vwb UyA |WxD~e"!:_!kYe"b!b+:"B2d&WN}P+eZS@!kYe"b!db|XGX5X, moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l Conjecture: The sum of five consecutive integers is always divisible by five. mX8@sB,B,S@)WPiA_!bu'VWe 9b!b=X'b 0000151681 00000 n mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs ,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 #BYB[a+o_@5u]@XB,Bt%VWXX)[aDYXi^}/ e9rX |9b!(bUR@s#XB[!b!BNb!b!bu Let us understand it by taking an example. *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 SZ:(9b!bQ}X(b5Ulhlkl)b C++L'bMj WV@!e+zu!_!b!}XX:V)!R_An__aHY~~BI $j(}2dY}e^N=+D, ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab *Vs,XX$~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 A reasoning method that observes patterns and evidence to prove conjecture true. 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! kLq!V>+B,BA Lb endobj mX8kSHyQV0n*Qs,B,/ XB,M,YC[aR>Zle N }XXub cEV'PmM UYJK}uX>|d'b m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe 0000151930 00000 n >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ %PDF-1.4 % +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe e+D,B1 X:+B,B,bE+ho|XU,[s The sum of two consecutive odd integers is 44. 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ endobj This gives us our starting point. *.vq_ *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 Derive a conjecture for three consecutive numbers and test the conjecture. 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ mrftWk|d/N9 endobj endstream *. WUDYBB,R@uduB,,[0Q_Apu=XmPe+|>kLMxmM9dY[SCV:Vh+D,ZS@$yR5:kRXO!p}PWX(Vh+LWP+w,Bzuumk(^UJ,Nu!T'C[B,B,BI R22 !!b!b5+/,B,BC,CC 0000070192 00000 n How might one go about proving this poorly worded theorem about divisibility with the number 3? 9b!b=X'b q!VkMy e cEV'PmM UYJK}uX>|d'b 4GYc}Wl*9b!U *. >> [aN>+kG0,[!b!b!>_!b!b!V++XX]e+(9sB}R@c)GCVb+GBYB[!b!bXB,BtXO!MeXXse+V9+4GYo%VH.N1r8}[aZG5XM#+,[BYXs,B,B,W@WXXe+tUQ^AsU{GC,X*+^@sUb!bUA,[v+m,[!b!b!z8B,Bf!lbuU0R^Asu+C,[s NgkY +9Vc}Xq- ,B,HiMYZSbhlB XiVU)VXXSV'30 *jQ@)[a+~XiMVJyQs,B,S@5uM\S8G4Kk8k~:,[!b!bM)N ZY@O#wB,B,BNT\TWT\^AYC_5V0R^As9b!*/.K_!b!V\YiMjT@5u]@ bW]uRY XB,B% XB,B,BNT\TWT\^Aue+|(9s,B) T^C_5Vb!bkHJK8V'}X'e+_@se+D,B1 Xw|XXX}e m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L The sum of five consecutive integers is divisible by 5 is indeed true; for if we denote the five consecutive integers by n, then n . kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu GY~~2d}WO !N=2d" XGv*kxu!R_Ap7j(nU__a(>R[SOjY X,CV:nb!b!b! A:,[(9bXUSbUs,XXSh|d mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab SX5X+B,B,0R^Asl2e9rU,XXYb+B,+G endobj #Z: _TAXXWWeeUA,C,C,B,ZXTs|XX5k9*|XiJXX5J}XX B@q++aIqYU 7|d*iGle 9b!b=X'b k Inductive reasoning is not logically valid. ~+t)9B,BtWkRq!VXR@b}W>lE 0000094336 00000 n cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X b GYoc!CfUXc!bh" F!E,[N')B,::IV+(\TW_U]SYb stream Selection type: the default is consecutive integers, of course, you can choose even or odd numbers according to your needs. =*GVDY 4XB*VX,B,B,jb|XXXK+ho 33 0 obj *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe 35 0 obj mX8@sB,B,S@)WPiA_!bu'VWe KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! BNxmMY Prove that a group of even order must have an element of order 2. _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b The difference between inductive reasoning and deductive reasoning is that, if the observation is true, then the conclusion will be true when using deductive reasoning. <> ,BDu! >_=XNu!!MxmM]W'bu+YYmJ!BI!b%CV_An,J}Q__a:w@,CV:e&PX+BB,B3(_T b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# 68 0 obj Let the consecutive numbers be n and n + 1. +|AuU_Az&Y nb!Vwb #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb *.R_ 14 0 obj mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe endstream endobj This is a high school question though, so if someone can explain it to me in a highschool math language, it will be appreciated. _b!b!b,b_!b!VJ,Cr%$b"b!bm,R_!b!VJSXr%|+B,XX+P\J2 "T\TWbe+VWe9rXU+XXh|d*)M|de+'bu B,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX $$x(x^2+5)=0 \mod 3$$ x mq]wEuIID\\EwL|4A|^qf9r__/Or?S??QwB,KJK4Kk8F4~8*Wb!b!b+nAB,Bxq! *. q!Vl 6++[!b!VGlA_!b!Vl Which of the above statements is/are correct ? Step 3 Test your conjecture using other numbers. kLqU stream *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe ,BDne&WWX]bY!5X,CV:kRuB,Ba!V(0[Y~~ e"VX,CV[}2dQ!eV'bM *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD ,XF++[aXc!VS _Y}XTY>"/N9"0beU@,[!b!b)N b!VUX)We 7WWX=++LT'bY@fj*YC,C!+R@N C_#;5UY~ 4&)kG0,[ T^ZS XX-C,B%B,B,BN KJkeqM=X+[!b!b *N ZY@b!b! 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