sum of five consecutive integers inductive reasoning sum of five consecutive integers inductive reasoning

What is continuous? endobj ^[aQX e m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L Write the inverse, converse, and contrapositive of the conditional statements below. mX8@sB,B,S@)WPiA_!bu'VWe KbRVX,X* VI-)GC,[abHY?le 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. 7WWXQ__a(Y7WSe2dMW!C,BBe_!b!b!CV_A SR^AsT'b&PyiM]'uWl:XXK;WX:X CONTACT; Email: Inventory Management Strategies Of Canadian Tire, How To Create 15 Minute Time Intervals In Excel, New Balance Indoor Nationals 2022 Standards. TB3WXXX+#WX+B,C,Cg\ 33XXXSWX'*'++a\ +b!bC@qMU+T?c|eXX8}XX+"22O_fJg\ 6gU+^Ob)UN,WBW cXB,BtX}XX+B,[X^)R_ ,[s ANSWER The sum of any two odd numbers is even. _)9Z:'bIb9rXBN5$~e T^ZSb,[C,[!b!~bE}e+D,ZU@)Br+L However, when using inductive reasoning, even though the statement is true, the conclusion wont necessarily be true. b9ER_9'b5 *. #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb Consecutive Integers can be written in the form: n, n + 1, n + 2, etc, where n is an integer. 8 0 obj Here, we have to consider only one counterexample to show this hypothesis false. 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Using the formula to calculate, the third integer is 17, so its 5 times is 5 * 17 = 85. 5 0 obj Each sum is even. endobj b"b!*. #T\TWT\@W' Sum of the smaller and twice the larger is -4. Show a counterexample for the given case to prove its conjecture false. #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b cEZ:Ps,XX$~eb!V{bUR@se+D/M\S 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 ):Ww+w,(!um;WYYhlX}X5:kRUp}P]WP}_+Vh+LWeXuN cB moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l q!Vl cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X '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 Do you agree that after your correction all we have to prove is $x^3+5x$ is always a multiple of $3$? kaqXb!b!BN mrk'b9B,JGC. 2021-04-26. X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d ,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 |d/N9 e+D,B1 X:+B,B,bE+ho|XU,[s endobj endobj stream 34 16060 XW+b!5u]@K 4X>l% T^\Syq!Bb!b ** #T\TWT\@W' So, the statements may not always be true in all cases when making the conjecture. K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& x+*00P A3S0i wI Yes I got it now. *.R_%VWe 35 B. <> +DHu!!kU!@Y,CVBY~Xg+B,XGY~#~mYO,B A reasoning method that observes patterns and evidence to prove conjecture true. Often inductive reasoning is referred to as the "Bottom-Up" approach as it uses evidence from specific scenarios to give generalized conclusions. I can prove deductively that they are divisible by $3$ but so far any combination I choose fails to prove the divisibility by $9$. *. 'bu mX8@sB,B,S@)WPiA_!bu'VWe _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 e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e !b!V: m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> e9rX%V\VS^A XB,M,Y>JmJGle Remember: Consecutive numbers are numbers that come after another in increasing order. *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 [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 $VRr%t% +Wb(jb!bC@}e*12B,B,Zv_!b!VJ,CjPUiJK&kc}XXz+MrbV+b5 We noticed that, starting with 3, every second number can be written as a sum of two consecutive numbers; starting with 6, every third number (6, 9, 12 . The type can be consecutive integers, consecutive even numbers or consecutive odd numbers. ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e _,9rkLib!V |d*)M.N B}W:XXKu_!b!b** *.*R_ k^q=X *.9r%_5Vs+K,Y>JJJ,Y?*W~q!VcB,B,B,BT\G_!b!VeT\^As9b5"g|XY"rXXc#~iW]#GVwe Identify your study strength and weaknesses. *.*b kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! 2 The product of three consecutive natural numbers can be equal to their sum. Make a test a conjecture about the sum of any three consecutive integers. |d/N9 *. hg(x+h)g(x)=cosx(h1cosh)sinx(hsinh). +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk WP,[a(w,Bsj(L_!b}:!!+R@N Kj*TT'bY@B,B:*VXp}P]WPM`e This decision is an example of inductive reasoning. endobj #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b 1 5, 1 6, 1 7, 1 8, 1 9. Over 10 million students from across the world are already learning smarter. mX8kSHyQV0n*Qs,B,/ XB,M,YC[aR>Zle Assume that the. 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b +e+|V+MIB,B,B}T+B,X^YB[aEy/-lAU,X'Sc!buG *.)ZYG_5Vs,B,z |deJ4)N9 In this question, the universal set, U, is the set of positive integers less than 20, and every set in this question is a subset of U. <> Describe how to sketch the fourth figure in the pattern. *. cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X ,BDne&WWX]bY!5X,CV:kRuB,Ba!V(0[Y~~ e"VX,CV[}2dQ!eV'bM So our conjecture is true for all even numbers. OyQ9VE}XGe+V(9s,B,Z9_!b!bjT@se+#}WYlBB,jbM"KqRVXA_!e |d/N9 K:QVX,[!b!bMKq!Vl Sum of Five Consecutive Integers Video. 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Click here to see ALL problems on Problems-with-consecutive-odd-even-integers Question 1098921 : If the sum of five consecutive even integers is t, then, in terms of t, what is the greatest integer? +e+|V+MIB,B,B}T+B,X^YB[aEy/-lAU,X'Sc!buG *Vs,XX$~e T^ZSb,YhlXU+[!b!BN!b!VWX8F)V9VEy!V+S@5zWX#~q!VXU+[aXBB,B X|XX{,[a~+t)9B,B?>+BGkC,[8l)b Did this satellite streak past the Hubble Space Telescope so close that it was out of focus? *.R_ #4GYc!,Xe!b!VX>|dPGV{b Here, the conclusion is drawn based on a statistical representation of the sample set. junio 16, 2022 . k^q=X 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ S: s,B,T\MB,B5$~e 4XB[a_ So, about 70% of doves are white. 0000128573 00000 n mrs7+9b!b Rw ,XF++[aXc!VS _Y}XTY>"/N9"0beU@,[!b!b)N b!VUX)We trailer << /Size 310 /Info 201 0 R /Root 204 0 R /Prev 480387 /ID[<3cd60dd519d6ab5c5d219b7bb7f06c6b><49b31575350be81b6df4257ea55a4316>] >> startxref 0 %%EOF 204 0 obj << /Type /Catalog /Pages 200 0 R /Metadata 202 0 R /AcroForm 206 0 R /Outlines 196 0 R /Names 207 0 R /OpenAction 205 0 R /ViewerPreferences << /HideToolbar true /HideMenubar true >> >> endobj 205 0 obj << /S /GoTo /D [ 210 0 R /FitH -32768 ] >> endobj 206 0 obj << /Fields [ 221 0 R 222 0 R 223 0 R 224 0 R 225 0 R 226 0 R 227 0 R 231 0 R 232 0 R 199 0 R 97 0 R 99 0 R 101 0 R 103 0 R 105 0 R 107 0 R 109 0 R 111 0 R 113 0 R 116 0 R 118 0 R 120 0 R 122 0 R 124 0 R 126 0 R 128 0 R 130 0 R 132 0 R 135 0 R 137 0 R 139 0 R 141 0 R 143 0 R 145 0 R 147 0 R 149 0 R 151 0 R 154 0 R 156 0 R 158 0 R 160 0 R 162 0 R 164 0 R 166 0 R 168 0 R 170 0 R 178 0 R 180 0 R 182 0 R 184 0 R 186 0 R 188 0 R 190 0 R 192 0 R 194 0 R 198 0 R ] /DR << /Font << /ZaDb 197 0 R /Helv 229 0 R >> /Encoding << /PDFDocEncoding 230 0 R >> >> /DA (/Helv 0 Tf 0 g ) >> endobj 207 0 obj << /JavaScript 208 0 R >> endobj 208 0 obj << /Names [ (disclosed)209 0 R ] >> endobj 209 0 obj << /S /JavaScript /JS (this.disclosed = true;\r\n) >> endobj 308 0 obj << /S 957 /O 1549 /V 1565 /Filter /FlateDecode /Length 309 0 R >> stream b=Ju_=`XXXXb_=XyMU|JXX+"22'+Msi$b"b!b5I4JJXAWzz:'Pqq!b!b!V_"b!VJ,C>Kg\ Disconnect between goals and daily tasksIs it me, or the industry? mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe #4GYc!,Xe!b!VX>|dPGV{b |d/N9 #BYB[a+o_@5u]@XB,Bt%VWXX)[aDYXi^}/ A conjecture is said to be true if it is true for all the cases and observations. KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! Think of it this way, each of the next 5 consecutive positive integers is 5 more than the corresponding first five integers. mB,B,R@cB,B,B,H,[+T\G_!bU9VEyQs,B1+9b!C,Y*GVXB[!b!b-,Ne+B,B,B,^^Aub! k~u!AuU_Ajj,*VX=N :>6'b9d9dEj(^[S n+Vzu!|J S: s,B,T\MB,B5$~e 4XB[a_ Conjecture: The sum of the first n odd positive integers is __?__. "T\TWbe+VWe9rXU+XXh|d*)M|de+'bu e9rX%V\VS^A XB,M,Y>JmJGle stream MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ _)9r_ "T\TWbe+VWe9rXU+XXh|d*)M|de+'bu #4GYcm }uZYcU(#B,Ye+'bu kN}Q__a}5X*0,BBet*eM,C!+R@5)ZFb!b!b=++LtVe&WWX]bY\eYe2dE&XB,B,B9GY~~nPb,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 b9ER_9'b5 d+We9rX/V"s,X.O TCbWVEBj,Ye Given an integer n, the task is to find whether n can be expressed as sum of five consecutive integer. ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl #Z: So, before that, we have to figure out two concepts: what is an integer? #Z: What is a word for the arcane equivalent of a monastery? 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UyA wVe SZ:(9b!bQ}X(b5Ulhlkl)b >Q@kWW _!b X_=dd:eY* Example: There are always white doves in the park. MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe ~iJ;WXX2B,BA X}+B,J'bbb!bUSbFJXXsNAub!b)9r%t%,)j? 2. !b!V: endobj cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu kLqU XW+b!5u]@K 4X>l% T^\Syq!Bb!b ** Conjecture: The next number will be 16, because 11+5=16. ,XF++[aXc!VS _Y}XTY>"/N9"0beU@,[!b!b)N b!VUX)We ?+B,XyQ9Vk::,XHJKsz|d*)N9"b!N'bu Case $2: x=3k+1$, then $x^2+2=9k^2+6k+1+2=3(3k^2+2k+1)$. *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD mX+#B8+ j,[eiXb #4GYc!,Xe!b!VX>|dPGV{b #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl *. K:QVX,[!b!bMKq!Vl x+*00P A3S0i w[ _)9r_ =*GVDY 4XB*VX,B,B,jb|XXXK+ho A:,[(9bXUSbUs,XXSh|d 'bu *.vq_ mrJyQszN9s,B,ZY@s#V^_%VSe(Vh+PQzlX'bujVb!bkHF+hc#VWm9b!C,YG eFe+_@1JVXyq!Vf+-+B,jQObuU0R^As+fU l*+]@s#+6b!0eV(Vx8S}UlBB,W@JS *.*b Andy and Belen made stars at the same time, making a total of 160 stars in 8 minutes. 7WWXQ__a(Y7WSe2dMW!C,BBzWXXu$*kWPM`eVWW=B,CV6TbYez:k(>+B,B,:XS5s+(\_A&j 67 0 obj 6XXX U}|5X*V;V>kLMxmM=K_!CCV:Vh+D,Z|u+*kxu!AuUBQ_!be+|(Vh+LT'b}e+'b9d9dEj(^[SECCVHY&XXb!b&X ?+B,XyQ9Vk::,XHJKsz|d*)N9"b!N'bu 40 0 obj e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX *.)ZYG_5Vs,B,z |deJ4)N9 KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! 70 0 obj #rk [a^A 4Xk|do+V@#VQVX!VWBB|X6++B,X]e+(kV+r_ 0000155651 00000 n e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX q!VkMy cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X 0000161836 00000 n <> KJkeqM=X+[!b!b *N ZY@b!b! |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb +9s,BG} 'Db}WXX8kiyWX"Qe b"b=XQ_!b!b!b}pV'bujB*eeXXM|uXXXhZB%JSXr%D,J4KXg\ WJ|eXX8S6bu !!VK4 m% XB,:+[!b!VG}[ This gives us our starting point. :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e *. Connect and share knowledge within a single location that is structured and easy to search. We 'bu 0000057079 00000 n Best study tips and tricks for your exams. mrJyQ1_ *.*b Sum of Integers Formula: S = n (a + l)/2. mX+#B8+ j,[eiXb 4&)kG0,[ T^ZS XX-C,B%B,B,BN 64 0 obj 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! 16060 k endstream True/False: What is the answer to the conjecture? Does ZnSO4 + H2 at high pressure reverses to Zn + H2SO4? >> WbY[SY_:_Yu!!MxmM]&P:k #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, ,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 endobj !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!VWe By using our site, you 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ ,|Bc^=dqXC,,Hmk mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab Now here is how I try to do it. W+,XX58kA=TY>" Step 1 1 of 3. Uu!zu@,C!UMxmM=tj(^]S$_]zBI!b!1 m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e 6++[!b!VGlA_!b!Vl 2d@b U!B,Bz35UY3>++LPW~ZC,BO2dWQWZmmR!0,B,BLbMU! XF+4GYkc!b5(O9e+,)M.nj_=#VQ~q!VKb!b:X #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! Nie wieder prokastinieren mit unseren Lernerinnerungen. b9ER_9'b5 #Z: XW+b!5u]@K 4X>l% T^\Syq!Bb!b ** SR^AsT'b&PyiM]'uWl:XXK;WX:X _WX B,B,@,C,C *.R_%VWe 6++[!b!VGlA_!b!Vl [ b65CVKi_9d9dN="b!^J endstream mrJyQ1_ cB b"b!. W'b:Xc!bk(^[SYgumWPV@{e+"bN :[}XC,^@$p}P]WP}u>llWPrF_! !Cumk(^]SmzC,[!b!bN :[}XC,__Ap}e+&;b!V65z B,}zBI!b!! Lb=y+|W,[aAuU_A endobj +++L'bi&WV@fj*Y2d^@{WXU&O~OXX[hY~ bbb!b!)z~a!b!b'bbbXbMMbVtWXXB,B!b!b=X_eeUA,C,C,B,Z=_5%V/,B,BC,C,CBbbMMbVtWXXB,B!b!b=X|bbbUuWMXr%D,BWXXWXXX+:X_!!V*|eXX+USbB,B,*.O922+r%,"++a\ g?b!b!b,9r%t%,!b!b!BN!VWeU+C,C +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe $Te m ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U Step 1: Find a rule by using few examples. B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX Answer (1 of 11): Let the smallest number of the five numbers be x. Nala is an orange cat and she purrs loudly. e+D,B,ZX@qb+B,B1 LbuU0R^Ab m% XB,:+[!b!VG}[ !}XXXGkfY}+(\T+(0Q_A{XHmWSe2dMW!C,BB _!b!b!CV_A 0000152370 00000 n endstream kLq!V To prove this conjecture true for all even numbers, lets take a general example for all even numbers. ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ >S?s|JJXR?B,B,B,W?)u.o*kaq!WX.O922B,m_5%+aXX5BB,Bxq++aIi ~+B,'bu which shows that n is sum of ve consecutive integers. ^[aQX e kLq!VH KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb s 4XB,,Y Sum of Multisets: The sum of two Then at least five computers are used by three or more students. That is, suppose that each number is either a multiple of $2$ or $3$. bbb!b!V_B,B,*.O92j=zk\ F B,B, *. A:,[(9bXUSbUs,XXSh|d 0000057583 00000 n _)9r_ 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! !*beXXMBl k^q=X endobj kaqXb!b!BN ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ _,9rkLib!V |d*)M.N B}W:XXKu_!b!b** e+D,B,ZX@qb+B,B1 LbuU0R^Ab 'bub!bC,B5T\TWb!Ve kLqU kLq!V>+B,BA Lb +C,C!++C!&!N b|XXXw+h <> Need to show that *. Conjecture: All quadrants of a circle are being filled with color in a clockwise direction. Observation: From the given pattern, we can see that every quadrant of a circle turns black one by one. [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 |d/N9 ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B |d/N9 e+D,B1 X:+B,B,bE+ho|XU,[s mrAU+XBF!pb5UlW>b 4IYB[aJ}XX+bEWXe+V9s k~u!R_ApV" 13 0 obj stream S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu cXB,BtX}XX+B,[X^)R_ +9s,BG} 'Db}WXX8kiyWX"Qe e+D,B,ZX@qb+B,B1 LbuU0R^Ab *.*b ,XF++[aXc!VS _Y}XTY>"/N9"0beU@,[!b!b)N b!VUX)We bWb!b!b!b!bWO V^S*.12B,B,}JXX+"22'+Msi$b"b!b5B,B,z&*'++ay%0B,B,B,B,z@N T\?c|eXX/j5UWbbEeeuWO VR)/Ir%D,B,j}XXLb)UN,WBW #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb VX>+kG0oGV4KhlXX{WXX)M|XUV@ce+tUA,XXY_}yyUq!b!Vz~d5Um#+S@e+"b!V>o_@QXVb!be+V9s,+Q5XM#+[9_=X>2 4IYB[a+o_@QXB,B,,[s *.F* 0000151995 00000 n .)ZbEe+V(9s,z__WyP]WPqq!s,B,,Y+W+MIZe+(Vh+D,5u]@X2B,ZRBB,Bx=UYo"ET+[a89b!b=XGQ(GBYB[a_ b9ER_9'b5 s 4Xc!b!F*b!TY>" kByQ9VEyUq!|+E,XX54KkYqU . #BYB[a+o_@5u]@XB,Bt%VWXX)[aDYXi^}/ ~+t)9B,BtWkRq!VXR@b}W>lE *.F* k =*GVDY 4XB*VX,B,B,jb|XXXK+ho #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl #Z: What is an all, always, every venn diagram? mrftWk|d/N9 what connection type is known as "always on"? K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& Although it looks a bit similar, there are still differences. cEV'PmM UYJK}uX>|d'b *.R_ C+|AuU_AB3je&PYguu=Xm8Q@5)B,::A!Y!e&+(\TWN :3BYHmkkufmM]W'jc XB,BC(_TR__aAuU_AB3+e&PYguuD6nN b!bR@zWoWe&+(\TWN :3BYHmkkufmM]W'vbQtsu!#,z(0Q_Apu!bee2dEj(^[S3kk:6 `u!#,z(0Q_Apu!bee2dEj(^[S3kk:G?+([@5)B,::A!F_O,C_aX~WP>+(\@$!u_! Yk(^[S3k WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d >> It's true when $x=0 \mod 3$. B,B, mX8kSHyQV0n*Qs,B,/ XB,M,YC[aR>Zle :e+We9+)kV+,XXW_9B,EQ~q!|d UXWXXe+VWe >zl2e9rX5kGVWXW,[aDY X}e+VXXcV !GY~~ 9b!b=X'b This reasoning is also used in scientific research by proving or contradicting a hypothesis. XW+b!5u]@K 4X>l% T^\Syq!Bb!b ** e9rX |9b!(bUR@s#XB[!b!BNb!b!bu *.vq_ .)ZbEe+V(9s,z__WyP]WPqq!s,B,,Y+W+MIZe+(Vh+D,5u]@X2B,ZRBB,Bx=UYo"ET+[a89b!b=XGQ(GBYB[a_ 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe #Z: #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl W~-e&WXC,C!}e2d-P!P_!b!sUb!b!ez(p+ #Z: ,[s *.R_%VWe 2d++Lu_+(\@5(C!k6YYTmmR_!b!b!b!be+L0A |d/N9 #4GYc!,Xe!b!VX>|dPGV{b ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ Write the following statement in if-then form We mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G JXX+6Jk #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe kLq!VH #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ k <> kLq!V>+B,BA Lb b 4IY?le WX+hl*+h:,XkaiC? 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! x+*00P A3S0i wm 'bu SZ:(9b!bQ}X(b5Ulhlkl)b s 4Xc!b!F*b!TY>" *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! KJkeqM=X+[!b!b *N ZY@b!b! #Z: +GYXr:J,Nu!VN ::N"B,B,B9 XWPB,GYB[aAuU@Xj|B,B'*MxmM=&PJ,Nu!T'jb}WXX5:AuU_A ++D,C!kMu!)M_h *UQ_!b!bm'|XGX5X, ,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_ mX+#B8+ j,[eiXb Therefore,k-2 + k-1 + k + k+1 + k+2= n5*k = nThe five numbers will be n/5 2, n/5 1, n/5, n/5 + 1, n/5 + 2. #T\TWT\@W' kLq++!b!b,O:'Pqy :X +9Vc}Xq- Some of the uses are mentioned below: Inductive reasoning is the main type of reasoning in academic studies. C,C,C,B1 X3}uXX5b}[?s|JJXR?8=B,B,B>S^R)/z+!b!D Determine whether the conjecture is true or false Dividing by 2 always produces a number less than the original number. _QAXX5l#22!b!b *9B,B,T@seeXU[b)UN,WBW b9ER_9'b5 wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d *.*b e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX Its 100% free. 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe We have to prove or disprove that the sum of these consecutive integers is divisible by 5 without leaving a remainder. 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Therefore, the sum of 5 consecutive odd numbers is equal to 5 times the third odd number. mrftWk|d/N9 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ 'b 4 0 obj V_keq!V++2!!VjJ_XXX 4XXXBJSXr%D,Bb_!b!b!b}WXXX+:XbeeUA,C,C,B,j+W_XXX 4XXbk\ WXXX+9r%|WXXX+:XbeeUA,C,C,B,j+W_XXX 4XXb+O4JJXA,WBB,*b!b!b!g\ u%|V'bu 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b stream *. x+*00P A3S0i w e9rX |9b!(bUR@s#XB[!b!BNb!b!bu *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* [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 *.R_%VWe ~WXUYc9(O j1_9rU,B,58[!_=X'#VX,[tWBB,BV!b=X uWX'VXA,XWe%q_=c+tQs,B58kVX+#+,[BYXUXWXXe+tUQ^AsWBXerkLq! 0000151012 00000 n 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 Then state the truth value d+We9rX/V"s,X.O TCbWVEBj,Ye Example #4: Look at the following patterns: 3 -4 = -12 The sum of 5 consecu 1. m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> KbRVX,X* VI-)GC,[abHY?le wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!VWe <> 'bub!bC,B5T\TWb!Ve 0000003418 00000 n *. The sum of two consecutive integers is 5, what are the integers? kByQ9VEyUq!|+E,XX54KkYqU MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe