sum of five consecutive integers inductive reasoning

0000057079 00000 n mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs endobj e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX 62 0 obj #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, [S@5&&PCCC,[kM I will be cubing, expanding and simplifying them endstream endobj 'bul"b *.N jb!VobUv_!V4&)Vh+P*)B,B!b! W+,XX58kA=TY>" Now, note that either $x$ is a multiple of $3$ or $(x^2+2)$ is a multiple of three. $$(3k - 1)((3k - 1)^2+5)=(3k - 1)(9k^2-6k+6)=0 \mod 3$$. #4GYcm }uZYcU(#B,Ye+'bu Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. 'b #4GYcm }uZYcU(#B,Ye+'bu d+We9rX/V"s,X.O TCbWVEBj,Ye ~+t)9B,BtWkRq!VXR@b}W>lE s 4Xc!b!F*b!TY>" +e+|V+MIB,B,B}T+B,X^YB[aEy/-lAU,X'Sc!buG s 4XB,,Y rev2023.3.3.43278. Think of it this way, each of the next 5 consecutive positive integers is 5 more than the corresponding first five integers. mrk'b9B,JGC. 2 is an even number but not composite, as it is a prime number. *. =*GVDY 4XB*VX,B,B,jb|XXXK+ho 9b!b=X'b [GYXr:+Zu!VN ::kb!bS_AjU_A{e+&+(\TW XikBuCYmkkrU'b e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX *.R_%VWe q!VkMy 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe SR^AsT'b&PyiM]'uWl:XXK;WX:X +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU mrftWk|d/N9 Given a number N, write a function to express N as sum of two or more consecutive positive numbers. 19 0 obj B,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb :e+We9+)kV+,XXW_9B,EQ~q!|d kLq!V>+B,BA Lb ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl e9rX%V\VS^A XB,M,Y>JmJGle [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e =*GVDY 4XB*VX,B,B,jb|XXXK+ho S"b!b A)9:(OR_ mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b e9rX |9b!(bUR@s#XB[!b!BNb!b!bu 4GYc}Wl*9b!U endobj *.9r%_5Vs+K,Y>JJJ,Y?*W~q!VcB,B,B,BT\G_!b!VeT\^As9b5"g|XY"rXXc#~iW]#GVwe XW+b!5u]@K 4X>l% T^\Syq!Bb!b ** 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: e+D,B,ZX@qb+B,B1 LbuU0R^Ab S: s,B,T\MB,B5$~e 4XB[a_ kLq!V>+B,BA Lb MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U Converse: If a number is a whole number, then it is a natural number :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e *.N1rV'b5GVDYB[aoiV} T^ZS T^@e+D,B,oQQpVVQs,XXU- 0000055164 00000 n B,B, +M,[; kByQ9VEyUq!|+E,XX54KkYqU b 4IY?le Where does this (supposedly) Gibson quote come from? endstream mrftWk|d/N9 +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU kLq!VH 6XXX b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B Find the next 5 terms in the sequence 38, 31, 24, 17, ___, ___, ___, ___, ___ . GV^Y?le 72 0 obj mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab A. +e+|V+MIB,B,B}T+B,X^YB[aEy/-lAU,X'Sc!buG Its 100% free. mrJyQszN9s,B,ZY@s#V^_%VSe(Vh+PQzlX'bujVb!bkHF+hc#VWm9b!C,YG eFe+_@1JVXyq!Vf+-+B,jQObuU0R^As+fU l*+]@s#+6b!0eV(Vx8S}UlBB,W@JS 16060 34 XF+4GYkc!b5(O9e+,)M.nj_=#VQ~q!VKb!b:X bbb!TbWjXXU\@suW"M4JJXA,WBCkEXXXo_}Xok~XXXXb+ZbEeeUA,C,C,DpA }X=h _b!b!b,b_!b!VJ,Cr%$b"b!bm,OR_!b!VJSXr%|+B,XX+P\G2 *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* 0000127387 00000 n vegan) just to try it, does this inconvenience the caterers and staff? 21 0 obj mrJyQ1_ log(x+2)log(x1)=log(x+2)log(x1)\frac { \log ( x + 2 ) } { \log ( x - 1 ) } = \log ( x + 2 ) - \log ( x - 1 ) s 4Xc!b!F*b!TY>" ,X'PyiMm+B,+G*/*/N }_ =b9dobU@{e+&PZG[|e+D,BE XGV'P>S*+BlD} XSFb 34 0 obj 'Db}WXX8kiyWX"Qe x+*00P A3S0i wv d+We9rX/V"s,X.O TCbWVEBj,Ye |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* _~WXXX)B,@w ~WXUYc9(O j1_9rU,B,58[!_=X'#VX,[tWBB,BV!b=X uWX'VXA,XWe%q_=c+tQs,B58kVX+#+,[BYXUXWXXe+tUQ^AsWBXerkLq! ,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 kByQ9VEyUq!|+E,XX54KkYqU Sum of N consecutive integers calculator start with first integer A. Conjecture: The product of two positive numbers is always greater than either number. cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X Save my name, email, and website in this browser for the next time I comment. JSXr%|0B,B,B,B,z@N T\?c|eXX5wj5UWbbEeeuWO VR)/Ir%D,B,;}XXLb)UN,WBW A simple example of inductive reasoning in mathematics. 'bub!bC,B5T\TWb!Ve mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab *.*b For example: What is the sum of 5 consecutive even numbers 60, 62, 64, 66 and 68? _WX B,B,22 !!b!b-6'bbb &VWmT9\ ] +JXXsZ+B,jbg\ ] KZ+B,jb!b!bmUbbbUWXXh+JSXr%D,B9-b!b53W%b!b5**eeXX+B,B 4XXXb)UN,WBW So, the statements may not always be true in all cases when making the conjecture. S: s,B,T\MB,B5$~e 4XB[a_ SR^AsT'b&PyiM]'uWl:XXK;WX:X *.*b k^q=X #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb ,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 Using the formula to calculate, the third even integer is 64, so its 5 times is 5 * 64 = 320, the answer is correct. If the statement is false, make the necessary change(s) to produce a true statement. The sum of any two consecutive integers is always odd. b"b!. 15,\,16,\,17,\,18,\,19 15, 16, 17, 18, 19. mrJyQ1_ #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, #rk [a^A 4Xk|do+V@#VQVX!VWBB|X6++B,X]e+(kV+r_ ,XF++[aXc!VS _Y}XTY>"/N9"0beU@,[!b!b)N b!VUX)We *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 6++[!b!VGlA_!b!Vl 0000094336 00000 n *. x mq]wEuIID\\EwL|4A|^qf9r__/Or?S??QwB,KJK4Kk8F4~8*Wb!b!b+nAB,Bxq! mX8kSHyQV0n*Qs,B,/ XB,M,YC[aR>Zle b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B q!VkMy 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. ^[aQX e Make a test a conjecture about the sum of any three consecutive integers. i_a:kYu!V@e+L(++B,7XS5s*,BD}&E}WN5+D,C!kxu)}e&&e vOy=}XXbbb!b!J S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ How might one go about proving this poorly worded theorem about divisibility with the number 3? !MU'b Example: 7 doves out of 10 I have seen are white. 5(n +2) If we divide this sum of any 5 consecutive integers by 5 we get: 5(n + 2) 5 . In addition to calculating the sum of 5 consecutive integers, you may also need to calculate the sum of 5 consecutive even numbers, or the sum of 5 consecutive odd numbers. *.*b _TAXXWWeeUA,C,C,B,ZXTs|XX5k9*|XiJXX5J}XX B@q++aIqYU [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 0000070192 00000 n 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e 4&)kG0,[ T^ZS XX-C,B%B,B,BN moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l 0000107763 00000 n =*GVDY 4XB*VX,B,B,jb|XXXK+ho +9s,BG} We Here, N represents an integer. Let us understand it by taking an example. :e+We9+)kV+,XXW_9B,EQ~q!|d 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. e+D,B1 X:+B,B,bE+ho|XU,[s +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk b W+,XX58kA=TY>" mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab 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 4 0 obj 0000057583 00000 n 3W%Xc+^@)B)u.j_bbU'bB,Bty!!!b!}Xb"b!*.Sy8 >+B,b!pe?dV)+ mX8kSHyQV0n*Qs,B,/ XB,M,YC[aR>Zle 13 0 obj #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe Consider 2 and 5. +9Vc}Xq- 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 KbRVX,X* VI-)GC,[abHY?le cB 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ $(x-1)^3+x^3+(x+1)^3=3x^3+6x=3(x^3+2x)=3x(x^2+2)$. cXB,BtX}XX+B,[X^)R_ Inductive reasoning is considered to be predictive rather than certain. _!!b&!0A,w+hn_VWX,CC({|e:,CVEY~Xu*~WuDXe+L can be written as a sum of four consecutive numbers. Create and find flashcards in record time. wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U + *Vs,XX$~e T^ZSb,YhlXU+[!b!BN!b!VWX8F)V9VEy!V+S@5zWX#~q!VXU+[aXBB,B X|XX{,[a~+t)9B,B?>+BGkC,[8l)b |d/N9 kByQ9VEyUq!|+E,XX54KkYqU m% XB,:+[!b!VG}[ 34 Conjecture: The sum of even numbers is an even number. A number is a neat number if the sum of the cubes of its digit equals the number. WP>+(_X/WeXuLukkY B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX mrAU+XBF!pb5UlW>b 4IYB[aJ}XX+bEWXe+V9s This reasoning is also used in scientific research by proving or contradicting a hypothesis. Given an integer n, the task is to find whether n can be expressed as sum of five consecutive integer. d+We9rX/V"s,X.O TCbWVEBj,Ye b 4IY?le KJkeqM=X+[!b!b *N ZY@b!b! mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab 'bul"b stream kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu mrk'b9B,JGC. "T\TWbe+VWe9rXU+XXh|d*)M|de+'bu But true observations by deductive reasoning will lead to true conjecture. moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l Complete the conjecture: The square of any negative number is ? ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B _b!b!b,b_!b!VJ,Cr%$b"b!bm,OR_!b!VJSXr%JO mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G mrk'b9B,JGC. &= x^3+x^3+3 x^2+3 x+1+x^3+6 x^2+12 x+8\\ ?l #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b 23 0 obj 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: +9s,BG} 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 'bul"b k~u!AuU_A4"_;GY~~z&Ya_YhYHmk ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ The sum of 5 consecutive positive integers = A. m% XB,:+[!b!VG}[ WX+hl*+h:,XkaiC? A. p}P]WPAuUOQ_ *Vs,XX$~e T^ZSb,YhlXU+[!b!BN!b!VWX8F)V9VEy!V+S@5zWX#~q!VXU+[aXBB,B X|XX{,[a~+t)9B,B?>+BGkC,[8l)b cB cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ * *.)ZYG_5Vs,B,z |deJ4)N9 mrk'b9B,JGC. 34 2. SX5X+B,B,0R^Asl2e9rU,XXYb+B,+G ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ Xg&PJ,CV:e&PvE_!b!b!#M`eV+h kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu |d/N9 <> What is the symbolic form of a contrapositive statement? *. ,[s NX~XXV'P>+(\CQ_Z+|(0Q@$!kY+2dN=2d" ) Show all of your work. 'bu b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# *Vs,XX$~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 Let us consider two integer numbers say -2 and -3. w *.vq_ 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe 34 RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)B* endobj ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ kaqXb!b!BN >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ 1 . b9ER_9'b5 Here we will understand what inductive reasoning is, compare it to related concepts, and discuss how we can give conclusions based on it. cXB,BtX}XX+B,[X^)R_ S cB YhYHmk kLq++!b!b,O:'Pqy For example, if you leave for work and it's raining outside, you reasonably assume that it will rain the whole way and decide to carry an umbrella. So, the next dove which comes will also be white. m%e+,RVX,B,B)B,B,B LbuU0+B"b . x mUwL .q)H;_swos?g??qc7GtW?w;vb!g+>b65u]@uu=XmDDu!jS 0000071114 00000 n s 4Xc!b!F*b!TY>" *.*b SX5X+B,B,0R^Asl2e9rU,XXYb+B,+G kByQ9VEyUq!|+E,XX54KkYqU +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!VWe Then use deductive reasoning to show that the conjecture is true. 'bul"b Wb}'XXC5u]@#U'b #4GYcm }uZYcU(#B,Ye+'bu n+VXQwD}!S@f e9rX%V\VS^A XB,M,Y>JmJGle 2.1 Use Inductive Reasoning Big Idea: To use INDUCTIVE REASONING in mathematics. [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X 7|d*iGle S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu 2 The product of three consecutive natural numbers can be equal to their sum. e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e s 4XB,,Y mrk'b9B,JGC. +9s,BG} WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d I appreciate it, We've added a "Necessary cookies only" option to the cookie consent popup. KJkeqM=X+[!b!b *N ZY@b!b! *. <> endstream MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie *. To prove this conjecture true for all even numbers, lets take a general example for all even numbers. GYoc!CfUXc!bh" F!E,[N')B,::IV+(\TW_U]SYb 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. #T\TWT\@W' +++L'bi&WV@fj*Y2d^@{WXU&O~OXX[hY~ W~-e&WXC,Cs!@Y,CVBY~Xb!b!ez(q_aKY~~ e"V:!}e2d-P!P_!b!b}XXDb=+|5_WWP_!bEhYY/eZ,C!+,BB, *.F* |d/N9 +GYc!b}>_!CV:!VN ::YYmMXX: b) Illustrate how the two algorithms you described in (a) can be used to find the spanning tree of a simple graph, using a graph of your choice with at least eight vertices and 15 edges. 0000002492 00000 n #BYB[a+o_@5u]@XB,Bt%VWXX)[aDYXi^}/ *.)ZYG_5Vs,B,z |deJ4)N9 +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe Notice that the sum of the five . =*GVDY 4XB*VX,B,B,jb|XXXK+ho # XGV'b_!b!BC+(\TW= .)ZbEe+V(9s,z__WyP]WPqq!s,B,,Y+W+MIZe+(Vh+D,5u]@X2B,ZRBB,Bx=UYo"ET+[a89b!b=XGQ(GBYB[a_ *.R_ 0000056695 00000 n KVX!VB,B5$VWe a concluding statement reached using inductive reasoning. 12 0 obj Suppose x and y are odd integers. B,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb !b!V: kLq!VH Through the above discussion, you should understand how to calculate the sum of 5 consecutive integers. Conjecture is the general conclusion which we reached by using induction reasoning. .)ZbEe+V(9s,z__WyP]WPqq!s,B,,Y+W+MIZe+(Vh+D,5u]@X2B,ZRBB,Bx=UYo"ET+[a89b!b=XGQ(GBYB[a_ cEZ:Ps,XX$~eb!V{bUR@se+D/M\S <> 2eYN5+D,jeT' *C $Pe+k =*GVDY 4XB*VX,B,B,jb|XXXK+ho ,X'PyiMm+B,+G*/*/N }_ +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb endstream #BYB[a+o_@5u]@XB,Bt%VWXX)[aDYXi^}/ ZknXX5vOy=}XXbbb!b!N KJkeqM=X+[!b!b *N ZY@b!b! XbbbUn++W5USbB,B,*.OB!lb)UN,WBW ?l stream Find two consecutive even integers whose sum is 126. wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U _)9Z:'bIb9rXBN5$~e T^ZSb,[C,[!b!~bE}e+D,ZU@)Br+L endobj #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb Conjecture: The product of two positive numbers is always greater than either number. 0. [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e KVX!VB,B5$VWe K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& m%e+,RVX,B,B)B,B,B LbuU0+B"b C,C,C,B1 (bMb"b!*.Sy'PqyVWX_bm-N[_!b!b!V)/MsiOyqY}XXXkIq=X?b!7 4XXXXch=&\ kNyB,kkqm&[B,B,B>S^R)/z+!b!J 0000003372 00000 n |d/N9 <> Example: 7 doves out of 10 in the U.S. are white. #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b x+*00P A3S0ih ~ *.vq_ x+*00P A3S0i w 32 0 obj How do I align things in the following tabular environment? The sum of two consecutive odd integers is 44. *. KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb +9_aX~~ bS@5:_Yu}e2d'!N=+D,k@XuWXO #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# kByQ9VEyUq!|+E,XX54KkYqU e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s endobj |d/N9 To subscribe to this RSS feed, copy and paste this URL into your RSS reader. #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b The conclusions obtained via inductive reasoning are only probable but not certain. UXWXXe+VWe >zl2e9rX5kGVWXW,[aDY X}e+VXXcV 0000072332 00000 n x+*00P A3S0i w These three situations are discussed separately below. mB,B,R@cB,B,B,H,[+T\G_!bU9VEyQs,B1+9b!C,Y*GVXB[!b!b-,Ne+B,B,B,^^Aub! 7|d*iGle That is, the sum of 5 consecutive even numbers is equal to 5 times the third even number. mX8kSHyQV0n*Qs,B,/ XB,M,YC[aR>Zle 2. mX8kSHyQV0n*Qs,B,/ XB,M,YC[aR>Zle 6XjH7|Xq++aIi B]byiK4XOb!bV'b@kLq! ,G b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B ?+B,XyQ9Vk::,XHJKsz|d*)N9"b!N'bu k^q=X S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu 6XXX Describe how to sketch the fourth figure in the pattern. kPy!!!uWmT9\ ] +JXXskWX mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs >> S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu U})E}e+e+|>kLMxmMszWUN= ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ #rk [a^A 4Xk|do+V@#VQVX!VWBB|X6++B,X]e+(kV+r_ *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* KVX!VB,B5$VWe kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! b Any statement that can be written in if-then form. mrftWk|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 Write the following statement in if-then form Derive a conjecture for three consecutive numbers and test the conjecture. 7O?o *,BD}!|e2dY5 X~Xb!b k #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 _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b a. '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 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 X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX #T\TWT\@W' ^[aQX e KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb 0000172261 00000 n *.F* *.)ZYG_5Vs,B,z |deJ4)N9 Example: All doves I have seen are white. DXX 6JzYs-m65292023591 - > > ()4~7 . |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s 'Db}WXX8kiyWX"Qe x mq]wEuIID\\EwL|4A|^qf9r__/Or?S??QwB,KJK4Kk8F4~8*Wb!b!b+nAB,Bxq! 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! 203 0 obj << /Linearized 1 /O 210 /H [ 3548 1385 ] /L 484577 /E 187344 /N 6 /T 480398 >> endobj xref 203 107 0000000016 00000 n mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu UyA kLq!V *'++a\ *.N jb!VobUv_!V4&)Vh+P*)B,B!b! X 6++[!b!VGlA_!b!Vl Conjecture Number 20 must be divisible by 5. :e+We9+)kV+,XXW_9B,EQ~q!|d S 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 ~+t)9B,BtWkRq!VXR@b}W>lE 0000151746 00000 n This is a high school question though, so if someone can explain it to me in a highschool math language, it will be appreciated. S"b!b A)9:(OR_ B,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb ~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 The sum of two consecutive integers is 5, what are the integers? #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, Get 247 customer support help when you place a homework help service order with us. 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 Let the consecutive numbers be n and n + 1. k^q=X mB&Juib5 "T\TWbe+VWe9rXU+XXh|d*)M|de+'bu endobj cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ m K:QVX,[!b!bMKq!Vl 17 0 obj 9b!b=X'b KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb 6++[!b!VGlA_!b!Vl MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ S So about 70% of doves in the U.S. are white. &4XS5s*,BDW@kWX5TY,CN!V@uWXQb!b=X_+B,@bMU! cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X 'Db}WXX8kiyWX"Qe *.vq_ mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe what connection type is known as "always on"? 27 0 obj e You can make the following conjecture. Select the smallest value of P that satisfies given conditions. 0000128573 00000 n 1.1 Inductive Reasoning filled in.notebook . #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe ,X'PyiMm+B,+G*/*/N }_ _)9r_ 0000172339 00000 n 0000067794 00000 n 'bu *. A:,[(9bXUSbUs,XXSh|d RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)B* KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe

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