e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e a = 2n + 1 and b = 2m+1, the definition of odd and even a+b = 2n + 1 + 2m + 1, the definition of sum. kLq!V nb!Vwb ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B Here, N represents an integer. e ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B 'b cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X KbRVX,X* VI-)GC,[abHY?le k m,b}lXGU'bM 0000128573 00000 n *.N jb!VobUv_!V4&)Vh+P*)B,B!b! e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e So if any one of the cases is false, the conjecture is considered false. .)ZbEe+V(9s,z__WyP]WPqq!s,B,,Y+W+MIZe+(Vh+D,5u]@X2B,ZRBB,Bx=UYo"ET+[a89b!b=XGQ(GBYB[a_ Connect and share knowledge within a single location that is structured and easy to search. +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU endobj K:QVX,[!b!bMKq!Vl WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b b:C;2dY}e?dVXX]_!b!bR!0Q_A{_|WWS__!bT'b=qY,CV_YY~5:kR C++L22d"2dYmbYBI!VWXXuh}Q__++0A,Bee2de2dE&X_!b!b!GY~~0D,B *.N1rV'b5GVDYB[aoiV} T^ZS T^@e+D,B,oQQpVVQs,XXU- Here are some examples of inductive reasoning that show how a conjecture is formed. *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD StudySmarter is commited to creating, free, high quality explainations, opening education to all. q!VkMy ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B Some of the uses are mentioned below: Inductive reasoning is the main type of reasoning in academic studies. *.F* I thought of doing a proof by contradiction. mB&Juib5 kLq!V>+B,BA Lb kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! 53 0 obj ,BDu! oN=2d" B_!b!b!#M`eV+h ,X'PyiMm+B,+G*/*/N }_ *.9r%_5Vs+K,Y>JJJ,Y?*W~q!VcB,B,B,BT\G_!b!VeT\^As9b5"g|XY"rXXc#~iW]#GVwe The sum of them is: n-2 + n-1 + n + n+1 + n+2 The -2 and +2 cancel out, the -1 and +1 cancel out, so you're just left with 5n. 'bub!bC,B5T\TWb!Ve k |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 *.F* This also has the advantage of working with various options to make a conjecture true. cEZ:Ps,XX$~eb!V{bUR@se+D/M\S >> *. >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ SR^AsT'b&PyiM]'uWl:XXK;WX:X S"b!b A)9:(OR_ cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X *. *.F* b 4IY?le :X]e+(9sBb!TYTWT\@c)G vaishnavikalesh4774 vaishnavikalesh4774 10.05.2019 S"b!b A)9:(OR_ *.)ZYG_5Vs,B,z |deJ4)N9 #4GYc!,Xe!b!VX>|dPGV{b *.*R_ mB&Juib5 VX>+kG0oGV4KhlXX{WXX)M|XUV@ce+tUA,XXY_}yyUq!b!Vz~d5Um#+S@e+"b!V>o_@QXVb!be+V9s,+Q5XM#+[9_=X>2 4IYB[a+o_@QXB,B,,[s 9b!b=X'b mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G kmR!e:fjk*,B,AA!b=XS5s+(\_A{WU'b&WXuC,CC!UW!0,B,zbI9d=+|W~~1e&XHu!!u_YY~ e!b!|XXLbMU!p}Q_++)0,2dEhYe2de+L(rzWXXe+LWe+B *.vq_ 364 0 obj <>stream :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e x mq]wEuIID\\EwL|4A|^qf9r__/Or?S??QwB,KJK4Kk8F4~8*Wb!b!b+nAB,Bxq! 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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 MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe b 0000053807 00000 n hg(x+h)g(x)=cosx(h1cosh)sinx(hsinh). WP,[a(w,Bsj(L_!b}:!!+R@N Kj*TT'bY@B,B:*VXp}P]WPM`e S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu *.9r%_5Vs+K,Y>JJJ,Y?*W~q!VcB,B,B,BT\G_!b!VeT\^As9b5"g|XY"rXXc#~iW]#GVwe "T\TWbe+VWe9rXU+XXh|d*)M|de+'bu S"b!b A)9:(OR_ 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb 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. *. kLq!VH ?+B,XyQ9Vk::,XHJKsz|d*)N9"b!N'bu m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L e+D,B,ZX@qb+B,B1 LbuU0R^Ab (By adding one more to the previous number you will get the next consecutive integer.) ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ *.F* Complete the conjecture: The square of any negative number is ? 'bu |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s stream mrAU+XBF!pb5UlW>b 4IYB[aJ}XX+bEWXe+V9s Determine whether the argument is an example of inductive reasoning (IR) or deductive reasoning (DR). 0000054170 00000 n #Z: UN=2dd_Y,C!J,BB,Z+B,BU:~+Weu5Y@kWW _!b X!%CVVY,C!J,BB%B,B $TeV+h mrftWk|d/N9 <> A. cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X Example: I have always seen doves during winter; so, I will probably see doves this winter. B,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb endstream #4GYc!,Xe!b!VX>|dPGV{b e WX+hl*+h:,XkaiC? 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: k ,X'PyiMm+B,+G*/*/N }_ b9ER_9'b5 endstream q!Vl S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu k Like even numbers, odd numbers are integers that are not divisible by 2. The sum of five consecutive integers is 100. find the third number. :X 'bu ~WXUYc9(O j1_9rU,B,58[!_=X'#VX,[tWBB,BV!b=X uWX'VXA,XWe%q_=c+tQs,B58kVX+#+,[BYXUXWXXe+tUQ^AsWBXerkLq! 5 0 obj mrftWk|d/N9 S MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie _)9r_ mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab :e+We9+)kV+,XXW_9B,EQ~q!|d _!!b&!0A,w+hn_VWX,CC({|e:,CVEY~Xu*~WuDXe+L B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!VWe 38 0 obj bWb!b!b!b!bWO V^S*.12B,B,}JXX+"22'+Msi$b"b!b5B,B,z&*'++ay%0B,B,B,B,z@N T\?c|eXX/j5UWbbEeeuWO VR)/Ir%D,B,j}XXLb)UN,WBW kLq!V 35 0 obj S *|eeU+C,B,zb!b!Vqy!!!}_!+a\ ] +JXXS|XXX+g\ ] K|eXX8SbbUWXXH_5%V/,B,BC,C,CB,W"bV m%e+,RVX,B,B)B,B,B LbuU0+B"b #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb 7|d*iGle . *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* +e+|V+MIB,B,B}T+B,X^YB[aEy/-lAU,X'Sc!buG bbb!TbWjXXU\@suW"M4JJXA,WBCkEXXXo_}Xok~XXXXb+ZbEeeUA,C,C,DpA }X=h endobj <> e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e *.9r%_5Vs+K,Y>JJJ,Y?*W~q!VcB,B,B,BT\G_!b!VeT\^As9b5"g|XY"rXXc#~iW]#GVwe m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L U}S*+ ?*'++a\ nsB,B,BN!VWO:XX_!bXXXX#|JJAC/ Consider the true statements Numbers ending with 0 and 5 are divisible by 5. kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU 7WWX=++LT'bY@fj*YC,C!+R@N C_#;5UY~ +9Vc}Xq- 0. 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[as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e 'bub!bC,B5T\TWb!Ve 0000136995 00000 n :X]e+(9sBb!TYTWT\@c)G mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe *Vs,XX$~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 :e+We9+)kV+,XXW_9B,EQ~q!|d 5. 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: [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e *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 mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ So, most of the doves are probably white. '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 cEV'PmM UYJK}uX>|d'b 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b 'bu ,[s 0000008844 00000 n 4GYc}Wl*9b!U A:,[(9bXUSbUs,XXSh|d XXX|uXXX22B,Bb!b!C,C,CU[b)UN,WBW Below is the implementation of this approach: Find last five digits of a given five digit number raised to power five, Count numbers up to N that cannot be expressed as sum of at least two consecutive positive integers, Check if a number can be expressed as a sum of consecutive numbers, Count primes that can be expressed as sum of two consecutive primes and 1, Count prime numbers that can be expressed as sum of consecutive prime numbers, Check if a given number can be expressed as pair-sum of sum of first X natural numbers, Check if a number can be expressed as sum two abundant numbers, Check if a number can be expressed as sum of two Perfect powers, Check if a number N can be expressed as the sum of powers of X or not, Check if a prime number can be expressed as sum of two Prime Numbers. OyQ9VE}XGe+V(9s,B,Z9_!b!bjT@se+#}WYlBB,jbM"KqRVXA_!e #4GYcm }uZYcU(#B,Ye+'bu For example: What is the sum of 5 consecutive even numbers 60, 62, 64, 66 and 68? <> stream *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD SZ:(9b!bQ}X(b5Ulhlkl)b The smaller of two consecutive integers is eight less than A straightforward word problem solved using an equation. The sum of two consecutive odd integers is 44. *. [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 Using the formula to calculate, the third even integer is 64, so its 5 times is 5 * 64 = 320, the answer is correct. k Observation: The product of the two numbers is positive. KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! <> b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: A:,[(9bXUSbUs,XXSh|d e e9rX |9b!(bUR@s#XB[!b!BNb!b!bu +GY~W~~1e"!kMu!S;|e2d:~+D XWXXuWX=:Wx [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 We k :X]e+(9sBb!TYTWT\@c)G k K:QVX,[!b!bMKq!Vl #BYB[a+o_@5u]@XB,Bt%VWXX)[aDYXi^}/ *.F* So, about 70% of doves are white. Make and test conjecture for the sum of two even numbers. k^q=X 4GYc}Wl*9b!U 6++[!b!VGlA_!b!Vl 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. An obtuse angle has a measure greater than 90 degrees, If an angle is obtuse, then is has a measure greater than 90 degrees, Write the following statement if if-then form ,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% XB,:+[!b!VG}[ |d/N9 6XjBwj?+WBWA X++b!V)/MsiOyiJK 71 0 obj 7|d*iGle b 4IY?le endobj R22 !!b!b5+/,B,BC,CC{BJSXr%D,Bb_!b!b!b}pV'buj-n q!Vl kLq!VH kLq!V K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& mrAU+XBF!pb5UlW>b 4IYB[aJ}XX+bEWXe+V9s 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ Hence, the smallest number is 43. rev2023.3.3.43278. 0000054781 00000 n =k4^`e!b=X+N=rFj(L_M% 6;}X5:kRUp}P]WP>+l ~+t)9B,BtWkRq!VXR@b}W>lE 0000094672 00000 n mrk'b9B,JGC. *.N jb!VobUv_!V4&)Vh+P*)B,B!b! 6XXX kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! 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 'Db}WXX8kiyWX"Qe #4GYc!,Xe!b!VX>|dPGV{b <> |d/N9 61 0 obj SR^AsT'b&PyiM]'uWl:XXK;WX:X ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ 6XXX |d/N9 :XIWSXWXE22 !!b!_vB,B,*.O90 Step 3: Test the conjecture for a particular set. e9rX |9b!(bUR@s#XB[!b!BNb!b!bu U}[(e+&+h K:'G b 4IY?le 48 0 obj KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb junio 16, 2022 . PE>Rh[=v:* ,i !.FU K?d)}[u8EZuMh}[7 ={.T8k8.xtbdco ^;?P> M m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L 37 0 obj $$x(x^2+5)=0 \mod 3$$ kByQ9VEyUq!|+E,XX54KkYqU 50 0 obj +MrbVkB,B_fiGkeq!V+(F,C,C +^u!_!b2d"+CV66)!bNkB5UY~e&:W~ZC,B2de2dE:WZmmRC_!b!V;:Xu_!b!k 16060 =*GVDY 4XB*VX,B,B,jb|XXXK+ho *.R_ <> 0000002769 00000 n Hence, the conjecture is false. mrAU+XBF!pb5UlW>b 4IYB[aJ}XX+bEWXe+V9s 0000057583 00000 n s 4Xc!b!F*b!TY>" e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! e Consecutive Integers can be written in the form: n, n + 1, n + 2, etc, where n is an integer. e9rX%V\VS^A XB,M,Y>JmJGle X2dU+(\TW__aX~We"V65oW,C!^@{e+D,Z+B,W'bMUp}P]Fb&WN}Q_!bEj(^[S;o{e2d X,BBBI*_aKY~~ Six consecutive integers equal 27 when added together. mrJyQ1_ ~WXUYc9(O j1_9rU,B,58[!_=X'#VX,[tWBB,BV!b=X uWX'VXA,XWe%q_=c+tQs,B58kVX+#+,[BYXUXWXXe+tUQ^AsWBXerkLq! kaqXb!b!BN b9ER_9'b5 ~+t)9B,BtWkRq!VXR@b}W>lE Caution: It is not always the case that the conjecture is true. >> ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl 6XXX +GYc!b}>_!CV:!VN ::YYmMXX: #rk [a^A 4Xk|do+V@#VQVX!VWBB|X6++B,X]e+(kV+r_ <> :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e KJkeqM=X+[!b!b *N ZY@b!b! KJkeqM=X+[!b!b *N ZY@b!b! .)ZbEe+V(9s,z__WyP]WPqq!s,B,,Y+W+MIZe+(Vh+D,5u]@X2B,ZRBB,Bx=UYo"ET+[a89b!b=XGQ(GBYB[a_ 4&)kG0,[ T^ZS XX-C,B%B,B,BN e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl #4GYc!,Xe!b!VX>|dPGV{b _)9r_ ?+B,XyQ9Vk::,XHJKsz|d*)N9"b!N'bu s 4XB,,Y 16060 mrftWk|d/N9 .)ZbEe+V(9s,z__WyP]WPqq!s,B,,Y+W+MIZe+(Vh+D,5u]@X2B,ZRBB,Bx=UYo"ET+[a89b!b=XGQ(GBYB[a_ 6XXX There are 10 consecutive nonzero positive integers. |d/N9 Test: We take three consecutive numbers 50,51,52. MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 nb!Vwb e+D,B1 X:+B,B,bE+ho|XU,[s ~WXUYc9(O j1_9rU,B,58[!_=X'#VX,[tWBB,BV!b=X uWX'VXA,XWe%q_=c+tQs,B58kVX+#+,[BYXUXWXXe+tUQ^AsWBXerkLq! 'bub!bC,B5T\TWb!Ve That is, suppose that each number is either a multiple of $2$ or $3$. mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe 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 0000003372 00000 n So, doves and geese are both of the same species. endobj *.*R_ 6XXX If a number is a natural number, then it is also a whole number, Inverse: IF a number is not a natural number, then it is not a whole number mrftWk|d/N9 'b #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, JSXw%XXXXX/6j5UWbbEe!V@4S^?JXXWXX$VRr%t% +k|!b!b!b!b!}b5u*O+C,B,B%D,Bx }XXbb=eJ_=XiJK&kI4JJXAWC,B,B,B,z4z:'Pqq!b!b!F_"b!VJ,C6Rz:OyiL"+!b!b!>_!b!b=XiJXY_=`XXXX#VW?k_ +^C_u%!VXXXi[OyiJK&k~@,B,z$*'++a_ X+KXXB~ T\^S*.12B,B,WBB,W]e!!!VSOyiJK&_h *. !PbXkf5XSWXQ__a}>+(\@kWX6YH2d@b U_!b!V;Dk{m k !*beXXMBl $$(3k + 1)((3k + 1)^2+5)=(3k + 1)(9k^2+6k+6)=0 \mod 3$$, ,XF++[aXc!VS _Y}XTY>"/N9"0beU@,[!b!b)N b!VUX)We OyQ9VE}XGe+V(9s,B,Z9_!b!bjT@se+#}WYlBB,jbM"KqRVXA_!e *. GV^Y?le 'bub!bC,B5T\TWb!Ve 0000005489 00000 n Conjecture: The sum of even numbers is an even number. 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe = 2n . [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 mrJyQ1_ *. e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX *. In addition to manual calculations, this page also provides a calculator for calculating the sum of 5 consecutive integers, so that you can get the sum of 5 consecutive integers faster. XF+4GYkc!b5(O9e+,)M.nj_=#VQ~q!VKb!b:X