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

K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b <> =*GVDY 4XB*VX,B,B,jb|XXXK+ho >S?s|JJXR?B,B,B,W?)u.*kaq! *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD d+We9rX/V"s,X.O TCbWVEBj,Ye A question of NUMBER THEORY and divisibility of 7. mB,B,R@cB,B,B,H,[+T\G_!bU9VEyQs,B1+9b!C,Y*GVXB[!b!b-,Ne+B,B,B,^^Aub! #Z: <> ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B endstream N b!bR@uF+B,VN}(Vf}QXX)3kkC!C,,[a:B}WXXp}P]RWX1e What's the difference between a power rail and a signal line. >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ cXB,BtX}XX+B,[X^)R_ mrftWk|d/N9 e9rX |9b!(bUR@s#XB[!b!BNb!b!bu ,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 +e+|V+MIB,B,B}T+B,X^YB[aEy/-lAU,X'Sc!buG }JT'bY@f HT{PG| A|0P#kQbJQ,UD`'DTb qZ*0 mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs *.F* *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe 4GYc}Wl*9b!U *.9r%_5Vs+K,Y>JJJ,Y?*W~q!VcB,B,B,BT\G_!b!VeT\^As9b5"g|XY"rXXc#~iW]#GVwe *.)ZYG_5Vs,B,z |deJ4)N9 UXWXXe+VWe >zl2e9rX5kGVWXW,[aDY X}e+VXXcV *. *.N jb!VobUv_!V4&)Vh+P*)B,B!b! >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ Sorry for the late reply. From the above, we can observe that the answer of all the sums is always an even number. %PDF-1.4 % 0000151746 00000 n KJkeqM=X+[!b!b *N ZY@b!b! :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e 'b b 4IY?le m b RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)B* endstream <> _~WXXX)B,@w Determine whether each equation is true or false. +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU N +B,:(Vh+LWP>+[aKYoc!b!&P~Wc5TYYYhlXBI!b%B,[a(V;V:kn}PXX]b9d9dEj(^[SC ^@5)B, endobj B,Bs&eWP>+ 'b,N Z=_=Xy!!!b!BbmwyN $}Xq++aIi B]byiK4#_!b!VB X+'+O922B,S@{B !b.O:'Pqyb!V)/MsiOyiJK+B,j^@8ke|b 4XXXXcVvW!B T\^S*.O:'uW_bm-N ZE_!OyiJKKS\?'|XXcV'b|X)O922B,S@B !b.O:'Pqy*9r%t%,)Z@ _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b Do}XXXXKJ,Ckaq=X?b!b!Vqy!!!b$_$++a\ kNyWXX3W%Xo *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 Hence, it is an even number, as it is a multiple of 2 and m+n is an integer. >> Inductive reasoning is used in academic studies, scientific research, and also in daily life. X>+kG0,[!b}X!*!b |X+B,B,,[aZ)=zle9rU,B,%|8g TY=?*W~q5!{}4&)Vh+D,B} XbqR^AYeE|X+F~+tQs,BJKy'b5 #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe endobj Prove that a group of even order must have an element of order 2. b 4IY?le 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ <> m% XB,:+[!b!VG}[ KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb endobj 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s U}S*+ +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU q!VkMy Conjecture The sum of any three consecutive integers is three times the second number. [aN>+kG0,[!b!b!>_!b!b!V++XX]e+(9sB}R@c)GCVb+GBYB[!b!bXB,BtXO!MeXXse+V9+4GYo%VH.N1r8}[aZG5XM#+,[BYXs,B,B,W@WXXe+tUQ^AsU{GC,X*+^@sUb!bUA,[v+m,[!b!b!z8B,Bf!lbuU0R^Asu+C,[s cB :e+We9+)kV+,XXW_9B,EQ~q!|d Conjecture: The next number will be 16, because 11+5=16. endstream k^q=X 3 0 obj S"b!b A)9:(OR_ by Sum of Consecutive Integers Word Problems. m% XB,:+[!b!VG}[ XXX|uXXX22B,Bb!b!C,C,CU[b)UN,WBW 0000003548 00000 n mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs UXWXXe+VWe >zl2e9rX5kGVWXW,[aDY X}e+VXXcV endobj mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs KVX!VB,B5$VWe 2021-04-26. Inductive reasoning is not logically valid. 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 31 0 obj S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s #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 ,Bn)*9b!b)N9 Lerne mit deinen Freunden und bleibe auf dem richtigen Kurs mit deinen persnlichen Lernstatistiken. #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ 2.1 Use Inductive Reasoning Big Idea: To use INDUCTIVE REASONING in mathematics. e kLq!V>+B,BA Lb W+,XX58kA=TY>" :X]e+(9sBb!TYTWT\@c)G cEZ:Ps,XX$~eb!V{bUR@se+D/M\S Here we will understand what inductive reasoning is, compare it to related concepts, and discuss how we can give conclusions based on it. kLqU 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. +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe K:'G #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe PDF 1.2 Inductive Reasoning - St. Joseph High School :X endobj WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d ,BDu! >_=XNu!!MxmM]W'bu+YYmJ!BI!b%CV_An,J}Q__a:w@,CV:e&PX+BB,B3(_T 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 #4GYc!,Xe!b!VX>|dPGV{b mB,B,R@cB,B,B,H,[+T\G_!bU9VEyQs,B1+9b!C,Y*GVXB[!b!b-,Ne+B,B,B,^^Aub! q!VkMy #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, S SZ:(9b!bQ}X(b5Ulhlkl)b *.R_%VWe mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs Algebra Forms of Linear Equations Applications Using Linear Models. *. <> As $3x(x^2+2)$ will have a multiple of three occurring once in the $3$, and once in either the $x$ or the $(x^2+2)$ term, we have that the sum of three consecutive cubes is a multiple of nine. ,;Reu7]Xd+! m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# endobj K:'G cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X B,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb ,Bn)*9b!b)N9 ,XF++[aXc!VS _Y}XTY>"/N9"0beU@,[!b!b)N b!VUX)We b *Vs,XX$~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 ^[aQX e KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb mrJyQ1_ +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU b. Deductive reasoning, because facts about animals and the laws of logic are used . :X]e+(9sBb!TYTWT\@c)G Using the formula to calculate, the third odd integer is 85, so its 5 times is 5 * 85= 425. *.F* e9rX |9b!(bUR@s#XB[!b!BNb!b!bu k 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! mX8kSHyQV0n*Qs,B,/ XB,M,YC[aR>Zle kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! For example, if you leave for work and it's raining outside, you reasonably assume that it will rain the whole way and decide to carry an umbrella. S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 e+D,B,ZX@qb+B,B1 LbuU0R^Ab #BYB[a+o_@5u]@XB,Bt%VWXX)[aDYXi^}/ +DHu!!kU!@Y,CVBY~Xg+B,XGY~#~mYO,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_ mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab 0000174791 00000 n +9Vc}Xq- 'Db}WXX8kiyWX"Qe 4GYc}Wl*9b!U XF+4GYkc!b5(O9e+,)M.nj_=#VQ~q!VKb!b:X x+*00P A3S0i w 1 1. b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# *.)ZYG_5Vs,B,z |deJ4)N9 *. <> 0000074662 00000 n ^[aQX e X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d q++aIi p*b!VBN!b/MsiS"2B,BA X+WXh_"b!*.SyT_bm-R_!b/N b!:OyqDU++C,B,T@}XkLqJ++!b!b,O:'PqyM +R@g_Yez:WX% ,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 I need to deductively prove that the sum of cubes of $3$ consecutive natural numbers is divisible by $9$. b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! m,b}lXGU'bM 0000071968 00000 n ,Bn)*9b!b)N9 =++DceZ+C!k~u!!MxuM!nb!BI!VAuU_AE,w+h sum of five consecutive integers inductive reasoning b"b=XQ_!b!b!b}pV'bujB*eeXXM|uXXXhZB%JSXr%D,J4KXg\ WJ|eXX8S6bu !!VK4 'bu KbRVX,X* VI-)GC,[abHY?le A reasoning method that observes patterns and evidence to prove conjecture true. ,XF++[aXc!VS _Y}XTY>"/N9"0beU@,[!b!b)N b!VUX)We UyA 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. #T\TWT\@W' Truth value: False, 0 SR^AsT'b&PyiM]'uWl:XXK;WX:X Are inductive and deductive the same type of reasoning? m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> 17 0 obj +D:_Yu!!+K6Y+e2dM+v%B9!nbMU!p}Q_aDYm)WW _!b'hY)2dYYmMXXb!k7*kWP(6eu4X~~ b"xb:u4,C!uT\YX5Xm!b!b(p}Q_\b&WXuC,CteYcB,B9jC!b=XS5s+(\_A{W s 4Xc!b!F*b!TY>" Z ?*'++a\ nsB,B,BN!VWO:XX_!bXXXX#|JJAC/ 0000004910 00000 n k 16060 *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe mB,B,R@cB,B,B,H,[+T\G_!bU9VEyQs,B1+9b!C,Y*GVXB[!b!b-,Ne+B,B,B,^^Aub! 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WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d b"b!V+B,B,ZY?s|JJX+C,B,B XBWXX2B,BWMXr%D,B)B,B,B3W%2B,B,ZY@) #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, kaqXb!b!BN 7|d*iGle SR^AsT'b&PyiM]'uWl:XXK;WX:X m% XB,:+[!b!VG}[ +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk e+D,B,ZX@qb+B,B1 LbuU0R^Ab '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 +R@Y/eZ,C X,BBBI*f,BD}Q_!bEj(^[S!C2d(zu!!++B,::kRJ}+l)0Q_A{WX Y!@YhY~Xi_!b!9 X2dU+(\TW_aKY~~ >> KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb 'b #4GYcm }uZYcU(#B,Ye+'bu 50 0 obj 'bu +C,C!++C!&!N b|XXXWe+B +9_aX~~ bS@5:_Yu}e2d'!N=+D,k@XuWXO Conjecture is the general conclusion which we reached by using induction reasoning. ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl 54 0 obj Hypothesis: Both numbers taken must be positive. ++D,C!kMu$VW3H2dUWXXB#B,M,C_aX~W(e} *. cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X Let n is sum of five consecutive integer of k 2, k-1, k, k + 1, k+2. UyA *.)ZYG_5Vs,B,z |deJ4)N9 #4GYc!,Xe!b!VX>|dPGV{b CC.912.G.CO.10 Prove theorems about triangles. mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe kaqXb!b!BN _)9r_ The answer to the above sum is an even number. Remember: Consecutive numbers are numbers that come after another in increasing order. endstream Conjecture: The sum of five consecutive integers is always divisible by five. #rk [a^A 4Xk|do+V@#VQVX!VWBB|X6++B,X]e+(kV+r_ x+*00P A3S0i wd s 4XB,,Y *. j XYYuu!b}lXB,BCe_!b=XSe+WP>+(\_A*_ 48 0 obj b"bu#VCXXX/-9r%_b!b!b,N T B| }XXbbb!b#VBJXXJ+ZXiJXX&bu !VJ|eXX8S Xj2k~$b"b!bm,O92z+MrbV+E_ B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX EXAMPLE 1 Make a Conjecture Complete the conjecture. Let $X$ stand for any natural number and let $X+1$ and $X+2$ stand for the two consecutive numbers. * 7O?o *,BD}!|e2dY5 X~Xb!b k B,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb W~GWXQ_!bYkKYg4XCe(Z* UXWXXe+VWe >zl2e9rX5kGVWXW,[aDY X}e+VXXcV Uu!b'}; XcI&Pzj(^[SC[ XBB,ZS@}XX:AuU_A SZ:(9b!bQ}X(b5Ulhlkl)b endstream nb!Vwb '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 >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ (By adding two more to the previous number you will get the next Q10. <> [aN>+kG0,[!b!b!>_!b!b!V++XX]e+(9sB}R@c)GCVb+GBYB[!b!bXB,BtXO!MeXXse+V9+4GYo%VH.N1r8}[aZG5XM#+,[BYXs,B,B,W@WXXe+tUQ^AsU{GC,X*+^@sUb!bUA,[v+m,[!b!b!z8B,Bf!lbuU0R^Asu+C,[s *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 'Db}WXX8kiyWX"Qe ,X'PyiMm+B,+G*/*/N }_ e+D,B1 X:+B,B,bE+ho|XU,[s #4GYcm }uZYcU(#B,Ye+'bu _!!b&!0A,w+hn_VWX,CC({|e:,CVEY~Xu*~WuDXe+L Providing a Foundation for Deductive Reasoning. d+We9rX/V"s,X.O TCbWVEBj,Ye GV^Y?le What is continuous? e9rX%V\VS^A XB,M,Y>JmJGle 7ojfY}+h |d/N9 S: s,B,T\MB,B5$~e 4XB[a_ For example, test that it works with . *.vq_ So, the next dove which comes will also be white. K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& d+We9rX/V"s,X.O TCbWVEBj,Ye ,XF++[aXc!VS _Y}XTY>"/N9"0beU@,[!b!b)N b!VUX)We *. #T\TWT\@W' *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 The sum of five consecutive integers is equal to the sum of the - Quora 6XXX stream 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b +9Vc}Xq- bbb!b!V_B,B,*.O92Z5k\ WXXX+9r%s%l+C,B,B Xzn #T\TWT\@W' b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU *.R_ 0000107763 00000 n s 4Xc!b!F*b!TY>" BNxmMY #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ U'bY@uduS-b!b p}P]WPAuU_A/GYoc!bS@r+rr^@Mxu![ XB,BCS_Ap}:%VK=#5ufmM=WYb9d * #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb cB #T\TWT\@W' B,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb Conjecture: The sum of the first n odd positive integers is __?__. +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk 0000073148 00000 n W'3ezWuB,C!B&XXT'P>+(:X, kByQ9VEyUq!|+E,XX54KkYqU WX+hl*+h:,XkaiC? mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe 66 0 obj Answer (1 of 10): "The sum of 5 consecutive integers equals the sum of the next 3 consecutive integers" This is the typical setup: n + (n+1) + (n+2) + (n+3) + (n+4) = (n+5) + (n+6) + (n+7) it's the least computationally taxing and it would have solved for the smallest number, and we can get . *.R_%VWe b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# We solved the question! |d/N9 +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!VWe MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie 2 The product of three consecutive natural numbers can be equal to their sum. +9Vc}Xq- EX . kV)!R_A{5WXT'b&WXzu!!(C4b U!5X~XWXXuWX=+ZC,B +e+|V+MIB,B,B}T+B,X^YB[aEy/-lAU,X'Sc!buG 'b vOy=}XXbbb!b!H +9Vc}Xq- x mUwL .q)H;_swos?g??qc7GtW?w;vb!g+>b65u]@uu=XmDDu!jS |d/N9 S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu kLq!V>+B,BA Lb |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb endstream _,9rkLib!V |d*)M.N B}W:XXKu_!b!b** OyQ9VE}XGe+V(9s,B,Z9_!b!bjT@se+#}WYlBB,jbM"KqRVXA_!e XF+4GYkc!b5(O9e+,)M.nj_=#VQ~q!VKb!b:X *. Find two consecutive positive integers whose product is 240. endobj _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b *.N jb!VobUv_!V4&)Vh+P*)B,B!b! !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!VWe SR^AsT'b&PyiM]'uWl:XXK;WX:X Write the following statement in if-then form kByQ9VEyUq!|+E,XX54KkYqU mrAU+XBF!pb5UlW>b 4IYB[aJ}XX+bEWXe+V9s 9b!b=X'b ,[s *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* *.R_%VWe e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e 6XXX e9rX |9b!(bUR@s#XB[!b!BNb!b!bu Xg&PJ,CV:e&PB,B,B9 GY}+We&Mx3}5XNu!!MxmM]W'b$'}XXXs(_T mX8kSHyQV0n*Qs,B,/ XB,M,YC[aR>Zle 'Db}WXX8kiyWX"Qe #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 }B,::B,_@{MxmM]W'b}l-e #Z: Consider two even numbers in the form: x=2m, y=2n, where x, y are even numbers and m, n are integers. mX+#B8+ j,[eiXb K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B cEZ:Ps,XX$~eb!V{bUR@se+D/M\S kLq!V>+B,BA Lb endobj M,C!+R@{J&&eY e"b!Ub!b UQ__a(_a]AZC,BBjeT'b&P66)eJ(F 7|d*iGle kLqU MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe N=2d" Yu!>+BB,ZT@uh}2dY_A{WWp}P]U'b} Y |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb My neighbors dog is also brown. Try It! *.9r%_5Vs+K,Y>JJJ,Y?*W~q!VcB,B,B,BT\G_!b!VeT\^As9b5"g|XY"rXXc#~iW]#GVwe *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* To subscribe to this RSS feed, copy and paste this URL into your RSS reader. stream 6XXX mrftWk|d/N9 VX>+kG0oGV4KhlXX{WXX)M|XUV@ce+tUA,XXY_}yyUq!b!Vz~d5Um#+S@e+"b!V>o_@QXVb!be+V9s,+Q5XM#+[9_=X>2 4IYB[a+o_@QXB,B,,[s mrJyQ1_ cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ N=2d" Yu!_!b!b-N :AuU_SW7N}Q__aAuU@1d}bhYHmkkCV@Ufe"b!BC+(\TWeu+CV(0Q_AN lmM~WUN=2d" Yu!_"bMp}P]5WV}Q__aAuU@5dV@{e2dEj(^[SB1+D,b!bS_AjY KJkeqM=X+[!b!b *N ZY@b!b! |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s ~WXUYc9(O j1_9rU,B,58[!_=X'#VX,[tWBB,BV!b=X uWX'VXA,XWe%q_=c+tQs,B58kVX+#+,[BYXUXWXXe+tUQ^AsWBXerkLq! mrk'b9B,JGC. &XbU3}5v+(\_A{WWpuM!5!}5X+N=2d" W'b_!b!B,CjY}+h w$TeVWWO @b6!kYHu!_!b!VX+B,B5 JYY~ stream +9Vc}Xq- 'b b1_YhYHmk Let us understand it by taking an example. *.N jb!VobUv_!V4&)Vh+P*)B,B!b! bbb!TbWjXXU\@suW"M4JJXA,WBCkEXXXo_}Xok~XXXXb+ZbEeeUA,C,C,DpA }X=h The sum of 5 consecutive integers is equal to 5 times the third integer. X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d *.vq_ e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX ?l &4XS5s*,BDW@kWX5TY,CN!V@uWXQb!b=X_+B,@bMU! b9ER_9'b5 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! _b!b!V^XXU\@seeuWJXD,WBW Explain who is correct and why. 7We+We R22 !!b!b5+/,B,BC,CC It is consistent with the above answer. Conjecture: The sum of three consecutive numbers is equal to three times the middle number of the given sum. Stop procrastinating with our smart planner features. Number 20 ends with 0. kLqU 6++[!b!VGlA_!b!Vl SX5X+B,B,0R^Asl2e9rU,XXYb+B,+G stream 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 moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l _,9rkLib!V |d*)M.N B}W:XXKu_!b!b** knXX5L #BI,WBW c++D,CCY,CV_YY~5:H_!b!bRC_a(_0,BB2dN=:a*_Y Arccsc Calculator Find the Exact Value of Inverse Cosecant, Arcsec Calculator Find the Exact Value of Inverse Secant, Arccot Calculator Find the Exact Value of Inverse Cotangent, Arctan Calculator Find the Exact Value of Inverse Tangent, Inverse Cosine Calculator Find The Exact Value of Arccos, Inverse Sine Calculator Find The Exact Value of Arcsin, Inverse Trigonometric Functions Calculator, Trigonometric Functions Conversion Calculator, Trig Calculator Find 6 Trigonometric Functions by Angles or Sides, Sum of Four Consecutive Integers Calculator, Sum of Three Consecutive Integers Calculator, Sum of Two Consecutive Integers Calculator. #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb #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 23303 XW+b!5u]@K 4X>l% T^\Syq!Bb!b ** 6Xb}kkq!~OyiJKKS\H2B,BA X+fN_!Gh'b *+b!V*.Sy'PqyMcW+WBWA X3OyiJKKS\K2B,BA X+ _!Gh'b5/+b!V*.Sy'PqyMW+WBWA X}OyiJKKS\N2B,BA X+zE_!Gh'b5kCXN T\@5u*R_!g\ ] KJ'bOyiJKKS\Q2B,BA X+tWC,C,C,B1 XMOCK_!z'PqyMT'_!Vkkq!Vb!bC,_R)/:7UkPq!B6BTy!!!b!B6r%D,X*.Sy'PqyM+_bm-N +B,Xu4S^?)unkPq!B6BTy!!!b!B6r%D,X*.Sy'PqyM+_bm-N +B,Xu4S^?)unkPq!B6BTy!!!b!B6r%D,X*.Sy'PqyM+_bm-N +B,Xu4S^?)unkPq!B6BTy!!!b!B6r%D,X*.Sy'PqyM+_bm-N +B,Xu4S^?)unkPq!B6BTy!!!b!B6r%D,X*.Sy'PqyM+_bm-N +B,Xu4S^?)unkPq!B6BTy!!!b!B6r%D,X*.Sy'PqyM+_bm-N +B,Xu4S^?)unkPq!B6BTy!!!b!B6r%D,X*.Sy'PqyM+_bm-N +B,Xu4S^?)unkPq!B6BTy!!!b!B6r%D,X*.Sy'PqyM+_bm-N +B,Xu4S^?)unkPq!B6BTy!!!b!B6r%D,X*.Sy'PqyM+_bm-N +B,Xu4S^?)unkPq!B6BTy!!!b!B6r%D,X*.Sy'PqyM+_bm-N +B,Xu4S^?)unkPq!B6BTy!!!b!B6I,WBB,S@5u*O*.S=}X+WBWA tbMXBN!b/MsiOyiJ[+C,B,T@8L4Iy!!!b!z,%+!b!b)O:'PqyBLq++aIi z"~8Qq!VKJ,C,BxX8F_ A conjecture is said to be true if it is true for all the cases and observations. 53 0 obj ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl &!t_}W[SSH+D,jXeb-X'bj *,BD[}P]WP:YmbYNw5e:,BBvUg?Y@udm k C,C,C,B1 4X|uXX5b}[?s|JJXR?8+B,B,B>S^R)/z+!b!H 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: cEV'PmM UYJK}uX>|d'b kByQ9VEyUq!|+E,XX54KkYqU KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! endobj mB,B,R@cB,B,B,H,[+T\G_!bU9VEyQs,B1+9b!C,Y*GVXB[!b!b-,Ne+B,B,B,^^Aub! Example: x2>x . #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl K:QVX,[!b!bMKq!Vl e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b So the conjecture is true for this given set. Example: All doves I have seen are white. ,XF++[aXc!VS _Y}XTY>"/N9"0beU@,[!b!b)N b!VUX)We 'bub!bC,B5T\TWb!Ve mrs7+9b!b Rw *. A:,[(9bXUSbUs,XXSh|d 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! ~iJWXX2B,BA Xm|XXhJ}J++!b!b,O:WXkOq!V22!b!b *N j+B,T@seeXU+W\ ] keyB,B=3W%X|XX{:Xu4!!VkPq!V_!b!C,C,C,BR_F|JJXX+Nb!b)9r%t%,)j+B,S@)B)un*|eXX 0000127093 00000 n endobj 20 0 obj We should always validate it, as it may have more than one hypothesis that fits the sample set. #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb mX8@sB,B,S@)WPiA_!bu'VWe 34 m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> *. +9s,BG} OyQ9VE}XGe+V(9s,B,Z9_!b!bjT@se+#}WYlBB,jbM"KqRVXA_!e endobj cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe *. 'Db}WXX8kiyWX"Qe #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe UXWXXe+VWe >zl2e9rX5kGVWXW,[aDY X}e+VXXcV ,!V!_!b=X+N=rFj(^]SOV"BIB,BshlD}e++Q@5&&P>u!k^N= "T\TWbe+VWe9rXU+XXh|d*)M|de+'bu e+D,B1 X:+B,B,bE+ho|XU,[s *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD cEZ:Ps,XX$~eb!V{bUR@se+D/M\S m%e+,RVX,B,B)B,B,B LbuU0+B"b 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ 0000070801 00000 n :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e /:X*0,BBee2de2dE&X_!b!b!GY~~0D,B 49 0 obj *. Using the formula to calculate, the third integer is 17, so its 5 times is 5 * 17 = 85. &=3x^{3}+9x^{2}+15x+9 \\ +0QLQ_h !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!VWe K:'G 0000068633 00000 n #T\TWT\@W' e 0000125414 00000 n WX+hl*+h:,XkaiC? stream We log(x+2)log(x1)=log(x+2)log(x1)\frac { \log ( x + 2 ) } { \log ( x - 1 ) } = \log ( x + 2 ) - \log ( x - 1 ) e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX 9b!b=X'b 0000002769 00000 n *.R_ ,[s ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B Describe how to create and solve. kxu!B,B,Z,J}Q_0,BB2dN=:d5|e2d:~+D XG *.N jb!VobUv_!V4&)Vh+P*)B,B!b! 2. 'bul"b 'Db}WXX8kiyWX"Qe moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l 34 KVX!VB,B5$VWe e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX |d/N9 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)B* _)9r_ mX+#B8+ j,[eiXb Step 1 1 of 3. k^q=X 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 ** S"b!b A)9:(OR_ _,9rkLib!V |d*)M.N B}W:XXKu_!b!b** m The sum of 5 consecutive integers is 105. what is the sum of - Quora 3. q!Vl ,X'PyiMm+B,+G*/*/N }_