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 . Conjecture: The next number will be 16, because 11+5=16. cEZ:Ps,XX$~eb!V{bUR@se+D/M\S ,XF++[aXc!VS _Y}XTY>"/N9"0beU@,[!b!b)N b!VUX)We . +9Vc}Xq- e+D,B1 X:+B,B,bE+ho|XU,[s #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, ,B,HiMYZSbhlB XiVU)VXXSV'30 *jQ@)[a+~XiMVJyQs,B,S@5uM\S8G4Kk8k~:,[!b!bM)N ZY@O#wB,B,BNT\TWT\^AYC_5V0R^As9b!*/.K_!b!V\YiMjT@5u]@ bW]uRY XB,B% XB,B,BNT\TWT\^Aue+|(9s,B) T^C_5Vb!bkHJK8V'}X'e+_@se+D,B1 Xw|XXX}e mrJyQszN9s,B,ZY@s#V^_%VSe(Vh+PQzlX'bujVb!bkHF+hc#VWm9b!C,YG eFe+_@1JVXyq!Vf+-+B,jQObuU0R^As+fU l*+]@s#+6b!0eV(Vx8S}UlBB,W@JS x+*00P A3S0i wv ~iJ;WXX2B,BA X}+B,J'bbb!bUSbFJXXsNAub!b)9r%t%,)j? endobj *.N jb!VobUv_!V4&)Vh+P*)B,B!b! moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l x+*00P A3S0ih ~ bbb!b!V_B,B,*.O92Z5k\ WXXX+9r%s%l+C,B,B Xzn SZ:(9b!bQ}X(b5Ulhlkl)b 0000005784 00000 n Example 2: The sum of an odd and an even number If an odd number and an even number are added, will the sum be an odd or an even number? 'bu Next step: The next pattern in this sequence will be: Next figure in sequence, Mouli Javia - StudySmarter Originals. *. _)9Z:'bIb9rXBN5$~e T^ZSb,[C,[!b!~bE}e+D,ZU@)Br+L b9ER_9'b5 #4GYc!,Xe!b!VX>|dPGV{b e9rX |9b!(bUR@s#XB[!b!BNb!b!bu ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ XH&P|e2d2d^@{WXAb+B,B5 JYY~ e9rX |9b!(bUR@s#XB[!b!BNb!b!bu mrk'b9B,JGC. ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L kLq!VH 2. mrJyQszN9s,B,ZY@s#V^_%VSe(Vh+PQzlX'bujVb!bkHF+hc#VWm9b!C,YG eFe+_@1JVXyq!Vf+-+B,jQObuU0R^As+fU l*+]@s#+6b!0eV(Vx8S}UlBB,W@JS Then the numbers are x, x + 1, x + 2, x + 3, and x + 4. kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu &=3x^{3}+9x^{2}+15x+9 \\ 3W%X+^@)B)u.nj_bbU'bB,Bty!!!b!}Xb"b!*.SyD b"b=XQ_!b!b!b}pV'bujB*eeXXM|uXXXhZB%JSXr%D,J4KXg\ WJ|eXX8S6bu !!VK4 UN=!;khY,CVX~X"B,!5X~Weuh mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab SR^AsT'b&PyiM]'uWl:XXK;WX:X ~+t)9B,BtWkRq!VXR@b}W>lE kaqXb!b!BN *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* 0000074950 00000 n cEV'PmM UYJK}uX>|d'b MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie endstream Make a test a conjecture about the sum of any three consecutive integers. #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe KJkeqM=X+[!b!b *N ZY@b!b! ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B b"bygXXXW XXXUbYK&kcyXqV!k6*'++a\ C,C,C,B1 4X{}uXX5b}[?s|JJXR?8&OiKJ,C,BxX8?X5 4XXXXch~JSXX5b_bm-NWXXKS\?S^O_u%!b!VS@m#_"b!*.Sy'Pq}XUR?s|JJXR?8: _!b!b!)/MsiOy=}XXXkIqu#B,**O922B,S@]_"b!*.Sy'Pq}XUR?s|JJXR?8B,B,BS^R)/z+!b!@ mX8@sB,B,S@)WPiA_!bu'VWe 21 0 obj kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! be_@{e2dEj(^]SuW'bM The type can be consecutive integers, consecutive even numbers or consecutive odd numbers. ?l #4GYc!,Xe!b!VX>|dPGV{b :e+We9+)kV+,XXW_9B,EQ~q!|d :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e Inductive reasoning sequence example, Mouli Javia - StudySmarter Originals. ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe It is consistent with the above answer. 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: #T\TWT\@2z(>RZS>vuiW>je+'b,N Z_!b!B Lb endobj d+We9rX/V"s,X.O TCbWVEBj,Ye ?l C++L22d"2dYmbYBI!VWXXuh}Q__++0A,Bee2de2dE&X_!b!b!GY~~0D,B *.9r%_5Vs+K,Y>JJJ,Y?*W~q!VcB,B,B,BT\G_!b!VeT\^As9b5"g|XY"rXXc#~iW]#GVwe *.N jb!VobUv_!V4&)Vh+P*)B,B!b! ZknXX5F[B,B,B,BS^O_u%!VXXXX8g?7XXsh+F_&*'++a\ kNywWXXcg\ ] KJg b!b!BN!b+B,C,C,B,ZX@B,B,T@seeX/%|JJX+WBWBB,ZY@]b!b!+WBWiJ7|XX58SX2'P7b+B,BA 4XXXUNWXb!b!BN!b+B,C,C,B,ZX@>_!b!b *O922BbWr%t%D,B TE_!b!b)9r%t%,)0>+B,B1 XB,_O_u%!VXXXX8R'bbb!5b}Wr%t%D,B TE_!b!b)9r%t%,) +B,B1 XB,_O_u%!VXXXX8^I * XXX|uXXX22B,Bb!b!C,C,C%z+MrbVWX wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U *.R_ 0000084731 00000 n #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl e mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> What sort of strategies would a medieval military use against a fantasy giant? 0000072355 00000 n G50j*aT <> Example: 7 doves out of 10 I have seen are white. +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> W'b:Xc!bk(^[SYgumWPV@{e+"bN :[}XC,^@$p}P]WP}u>llWPrF_! !Cumk(^]SmzC,[!b!bN :[}XC,__Ap}e+&;b!V65z B,}zBI!b!! Lb=y+|W,[aAuU_A *.N jb!VobUv_!V4&)Vh+P*)B,B!b! 61 0 obj e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e KbRVX,X* VI-)GC,[abHY?le l|X cXB,BtX}XX+B,[X^)R_ XXXfq+)ZbEeeUA,C,C,LiJK&kcy_ki5XiJX_!b!VVP+_C_u%!VXXX _fJg\ 6P+^Ob)UN,WBW +9Vc}Xq- Inductive reasoning uses previous examples and patterns to form a conjecture. #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 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>+(\@$!u_! Yk(^[S3k mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab OyQ9VE}XGe+V(9s,B,Z9_!b!bjT@se+#}WYlBB,jbM"KqRVXA_!e mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs Now we test this conjecture on another sequence to consider if the derived conclusion is in fact true for all consecutive numbers. kLq!V>+B,BA Lb kLq!V>+B,BA Lb This reasoning gives a chance to explore the hypothesis in a wider field. *.R_ KVX!VB,B5$VWe That is 34 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ s 4XB,,Y +C,C!++C!&!N b|XXXWe+B The five consecutive integers are 15,16,17,18,19 Explanation: To identify five consecutive integers we begin by giving them each a variable expression 1st = x 2nd = x + 1 3rd = x + 2 4th = x + 3 5th = x + 4 Now we set these equal to a sum of 85 x + x + 1 + x + 2 +x +3 +x +4 = 85 5x +10 = 85 5x+10 10 = 85 10 5x = 75 5x 5 = 75 5 x = 15 Then sketch Everything you need for your studies in one place. 14 0 obj 'bub!bC,B5T\TWb!Ve +9Vc}Xq- endobj 6++[!b!VGlA_!b!Vl Specific observation. 15717 mrJyQ1_ _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)B* Express the fraction 164 using negative exponent. Inductive Reasoning Inductive Reasoning Inductive Reasoning Calculus Absolute Maxima and Minima Absolute and Conditional Convergence Accumulation Function Accumulation Problems Algebraic Functions Alternating Series Antiderivatives Application of Derivatives Approximating Areas Arc Length of a Curve Area Between Two Curves Arithmetic Series StudySmarter is commited to creating, free, high quality explainations, opening education to all. 0000054543 00000 n KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! *. mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs s 4Xc!b!F*b!TY>" +MrbV}XXX+WWV'buq+E,C,C 'Db}WXX8kiyWX"Qe _)9Z:'bIb9rXBN5$~e T^ZSb,[C,[!b!~bE}e+D,ZU@)Br+L *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* +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 m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe *.*b 42 0 obj Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. ,B&PC2d(zu!!++B,::kRJ}+l)0Q_A{WXCVW,Ce^N=2d"b}XXT'bMUp}P]5W~-e&+h 7UW|z>kLMxmM9d+, XB[!b!J Make a conjecture about the next number in the given sequence. kLq!V>+B,BA Lb True. mrftWk|d/N9 moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l k What is the third number? wV__a(>R[S3}e2dN=2d" XGvB,ZW@5)WP>+(J[WW=++D!zYHu!!N :|5WYX&X Inductive reasoning, because a pattern is used to reach the conclusion. >X@{MxmM]W'|bWse+(VXX[V_!b!b!Te 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. UXWXXe+VWe >zl2e9rX5kGVWXW,[aDY X}e+VXXcV #4GYc!,Xe!b!VX>|dPGV{b 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b 51 0 obj W'3ezWuB,C!B&XXT'P>+(:X, 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. b 4IY?le +e+|V+MIB,B,B}T+B,X^YB[aEy/-lAU,X'Sc!buG b mrk'b9B,JGC. #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ s 4Xc!b!F*b!TY>" 37 0 obj k^q=X 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ let a and b be odd numbers. +9s,BG} ?l WbY[SY_:_Yu!!MxmM]&P:k )#j(^[S MxmM]W'FN b!bR@zg_ B,B,R@B,B,BI mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe 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: *. <> XGV'P|;b!VXYYumh^C0U@5)B,::&e_!b!b! 44 0 obj *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d +DHu!!k!@Y,CVBY~Xb!b!ez(p0+ Find a counterexample for: All even numbers are composite. kLqX_++!b!b,O:'PqywWX%3W%X[+B,B,ZX?)u.)+b!b-)Non m%e+,RVX,B,B)B,B,B LbuU0+B"b 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b 4&)kG0,[ T^ZS XX-C,B%B,B,BN So the conjecture is true for this given set. The sum of 5 consecutive integers can be 100. WSB3WXXX+WX+B,C,Cr%$b"b!bm,R_!b!VJSXr%D/ !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!VWe *. #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b m Step 1: Find the pattern between these groups. C. Prove using deductive reasoning the following conjectures. x mq]wEuIID\\EwL|4A|^qf9r__/Or?S??QwB,KJK4Kk8F4~8*Wb!b!b+nAB,Bxq! 'bub!bCHyUyWPqyP]WTyQs,XXSuWX4Kk4V+N9"b!BNB,BxXAuU^AT\TWb+ho" X+GVc!bIJK4k8|#+V@se+D,B1 X|XXB,[+U^Ase+tUQ^A5X+krXXJK4Kk+N9 9b!b=X'b sum of five consecutive integers inductive reasoning. *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l Truth value: True. 47 0 obj d+We9rX/V"s,X.O TCbWVEBj,Ye b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# endobj ,X'PyiMm+B,+G*/*/N }_ SZ:(9b!bQ}X(b5Ulhlkl)b 'b m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X
sum of five consecutive integers inductive reasoning
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