sum of five consecutive integers inductive reasoning

|d/N9 [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d Inductive reasoning, because a pattern is used to reach the conclusion. 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! 4&)kG0,[ T^ZS XX-C,B%B,B,BN That is, suppose that each number is either a multiple of $2$ or $3$. 'bul"b cB 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b endstream #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 +++LWe!!+R@fj*Y2d^@{WX5Xb!b!bMR!0Q_A&j This type of reasoning forms a causal connection between evidence and hypothesis. mrs7+9b!b Rw x mq]wEuIID\\EwL|4A|^qf9r__/Or?S??QwB,KJK4Kk8F4~8*Wb!b!b+nAB,Bxq! rev2023.3.3.43278. b9ER_9'b5 20 C. 12 D. 30 E. 56 16. . e9rX |9b!(bUR@s#XB[!b!BNb!b!bu K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ =*GVDY 4XB*VX,B,B,jb|XXXK+ho 'bu If the conjecture is FALSE, give a counter example. cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X :X]e+(9sBb!TYTWT\@c)G Given a number N, write a function to express N as sum of two or more consecutive positive numbers. 4GYc}Wl*9b!U 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,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B This is a high school question though, so if someone can explain it to me in a highschool math language, it will be appreciated. XW+b!5u]@K 4X>l% T^\Syq!Bb!b ** +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 *.*b ~+t)9B,BtWkRq!VXR@b}W>lE e9rX%V\VS^A XB,M,Y>JmJGle ^[aQX e SX5X+B,B,0R^Asl2e9rU,XXYb+B,+G 5 0 obj k~u!AuU_Abe+|(Vh+LT'b6'b9d9dEj(^[S x_9de+|:kRXuH mB,B,R@cB,B,B,H,[+T\G_!bU9VEyQs,B1+9b!C,Y*GVXB[!b!b-,Ne+B,B,B,^^Aub! *.9r%_5Vs+K,Y>JJJ,Y?*W~q!VcB,B,B,BT\G_!b!VeT\^As9b5"g|XY"rXXc#~iW]#GVwe So if any one of the cases is false, the conjecture is considered false. 60 + 62 + 64 + 66 + 68 = 320. mX+#B8+ j,[eiXb *.R_ <> cEZ:Ps,XX$~eb!V{bUR@se+D/M\S b9ER_9'b5 mrftWk|d/N9 SX5X+B,B,0R^Asl2e9rU,XXYb+B,+G Consider groups of three consecutive numbers. mrs7+9b!b Rw ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ UyA WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d <> m%e+,RVX,B,B)B,B,B LbuU0+B"b cEV'PmM UYJK}uX>|d'b 'Db}WXX8kiyWX"Qe 0000128987 00000 n MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie e [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 MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie - The product of two odd numbers is odd. q++aIi @*b!VBN!b/MsiR"2B,BA X+WXhg_"b!*.SyR_bm-R_!b/N b!:Oyq,U++C,B,T@}XkLq2++!b!b,O:'Pqy5 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 kLq!VH mrs7+9b!b Rw 0000003474 00000 n bbb!b!V_B,B,*.O92j=zk\ F b 4IY?le p}P]WPAuUOQ_ :X]e+(9sBb!TYTWT\@c)G ,[s 8 0 obj A:,[(9bXUSbUs,XXSh|d Inductive reasoning, because a pattern is used to reach the conclusion. *.F* 0000117497 00000 n x+*00P A3S0i w[ If yes, find the five consecutive integers, else print -1.Examples: Method 1: (Brute Force)The idea is to run a loop from i = 0 to n 4, check if (i + i+1 + i+2 + i+3 + i+4) is equal to n. Also, check if n is positive or negative and accordingly increment or decrement i by 1.Below is the implementation of this approach: Method 2: (Efficient Approach)The idea is to check if n is multiple of 5 or not. 'Db}WXX8kiyWX"Qe K:QVX,[!b!bMKq!Vl We >> 2eYN5+D,jeT' *C $Pe+k 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 kMuRVp7Vh+)Vh+L'b : >_!b9dzu!VXqb}WB[!b!BI!b5We ,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 endstream e9rX%V\VS^A XB,M,Y>JmJGle *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe Now we just have to prove $3|x$ or $3|x^2+2$. +++LtU}h Answer (1 of 8): \text{Suppose that the integers are $n-2, n-1, n, n+1, n+2$.} WP}_o$Te kLq++!b!b,O:'Pqy D:U!_;GY_+ZC,B *.R_ moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l cEV'PmM UYJK}uX>|d'b The case which shows the conjecture is false is called the counterexample for that conjecture. 'bub!bCHyUyWPqyP]WTyQs,XXSuWX4Kk4V+N9"b!BNB,BxXAuU^AT\TWb+ho" X+GVc!bIJK4k8|#+V@se+D,B1 X|XXB,[+U^Ase+tUQ^A5X+krXXJK4Kk+N9 *. 34 This reasoning has limited scope and, at times, provides inaccurate inferences. *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* #4GYc!,Xe!b!VX>|dPGV{b WX+hl*+h:,XkaiC? Although it looks a bit similar, there are still differences. The general unproven conclusion we reach using inductive reasoning is called a conjecture or hypothesis. mX8@sB,B,S@)WPiA_!bu'VWe *. 13 0 obj kLq!V SR^AsT'b&PyiM]'uWl:XXK;WX:X ,B,HiMYZSbhlB XiVU)VXXSV'30 *jQ@)[a+~XiMVJyQs,B,S@5uM\S8G4Kk8k~:,[!b!bM)N ZY@O#wB,B,BNT\TWT\^AYC_5V0R^As9b!*/.K_!b!V\YiMjT@5u]@ bW]uRY XB,B% XB,B,BNT\TWT\^Aue+|(9s,B) T^C_5Vb!bkHJK8V'}X'e+_@se+D,B1 Xw|XXX}e N +B,:(Vh+LWP>+[aKYoc!b!&P~Wc5TYYYhlXBI!b%B,[a(V;V:kn}PXX]b9d9dEj(^[SC ^@5)B, kLqU *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 Which of the following is not a type of inductive reasoning? 0000005489 00000 n #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ Let S be the number of perfect squares among the integers from 1 to 20136. 'bu b 4IY?le *.F* cEV'PmM UYJK}uX>|d'b mrJyQ1_ kLq!V>+B,BA Lb X2dU+(\TWu__aX~We"V65u;}e2d X,BB+B,W'bMUp}P]RW~~!bS_A{WX9C[2dYC,C_!b!_!b!V:kRJ}++ e _)9Z:'bIb9rXBN5$~e T^ZSb,[C,[!b!~bE}e+D,ZU@)Br+L 'bul"b #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b nb!Vwb Thus, answer choice C A+25 is correct. *.F* mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab 4GYc}Wl*9b!U VXT9\ ] +JX=_!,9*!m_!+B,C,C Consider the true statements Numbers ending with 0 and 5 are divisible by 5. ^[aQX e stream kKu!Qb!z&*VXp}P]WP>e+|(>R[SY[!k~u!VN ::"BI!b!1b! 4GYc}Wl*9b!U k [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 stream Prime numbers only have two factors, 1 and itself, If prime numbers only have 2 factors, then they are 1 and itself. #4GYc!,Xe!b!VX>|dPGV{b |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s q!Vl ,Bn)*9b!b)N9 kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu ,[s q!VkMy *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 K:QVX,[!b!bMKq!Vl MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe 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 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. #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b The answer to the above sum is an even number. W+,XX58kA=TY>" e MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe .)ZbEe+V(9s,z__WyP]WPqq!s,B,,Y+W+MIZe+(Vh+D,5u]@X2B,ZRBB,Bx=UYo"ET+[a89b!b=XGQ(GBYB[a_ Since the middle integer, n, . +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk mX8@sB,B,S@)WPiA_!bu'VWe ~WXUYc9(O j1_9rU,B,58[!_=X'#VX,[tWBB,BV!b=X uWX'VXA,XWe%q_=c+tQs,B58kVX+#+,[BYXUXWXXe+tUQ^AsWBXerkLq! mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab q!Vl what connection type is known as "always on"? b mX8kSHyQV0n*Qs,B,/ XB,M,YC[aR>Zle bbb!6bTX?JXX+ B'+MrbV+N B,jb!b-)9I_"O+C,B,B @bXC*eeX+_C?3XXXh *.N1rV'b5GVDYB[aoiV} T^ZS T^@e+D,B,oQQpVVQs,XXU- _)9r_ s 4XB,,Y *.vq_ #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b &=3x^{3}+9x^{2}+15x+9 \\ cEV'PmM UYJK}uX>|d'b n&B,B,ZS@uWXp70,BD}!|e >_YYW'b"b@ Disconnect between goals and daily tasksIs it me, or the industry? 'bul"b X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d endobj *. 'Db}WXX8kiyWX"Qe 9b!b=X'b mrs7+9b!b Rw *.*R_ *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 K:'G W+,XX58kA=TY>" <> =W~GWXQ_!bYkh~SY!kYe"b!Fb}WuDXe+L S kLq!VH W+,XX58kA=TY>" 0000084754 00000 n ~+t)9B,BtWkRq!VXR@b}W>lE . ?l +0QLQ_h 4&)kG0,[ T^ZS XX-C,B%B,B,BN +JXXXXWh1zk\ WXXX+9r%%keq!VM 'Db}WXX8kiyWX"Qe endstream #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl 'b kaqXb!b!BN $$(3k + 1)((3k + 1)^2+5)=(3k + 1)(9k^2+6k+6)=0 \mod 3$$, e9rX |9b!(bUR@s#XB[!b!BNb!b!bu e9rX |9b!(bUR@s#XB[!b!BNb!b!bu 6;}X5:kRUp}P]WP>+l *. *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD Find two consecutive positive integers whose product is 240. [S@5&&PCCC,[kM *.F* W~-e&WXC,C!}e2d-P!P_!b!sUb!b!ez(p+ XXX|uXXX22B,Bb!b!C,C,C%z+MrbVWX Describe how to sketch the fourth figure in the pattern. endobj Upload unlimited documents and save them online. GY~~2d}WO !N=2d" XGv*kxu!R_Ap7j(nU__a(>R[SOjY X,CV:nb!b!b! This gives us our starting point. We MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie endobj MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe 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 4&)kG0,[ T^ZS XX-C,B%B,B,BN ?l Truth value: True. 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 ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl e+|(9s,BrXG*/_jYiM+Vx8SXb!b)N b!VEyP]7VJyQs,X X}|uXc!VS _YiuqY]-*GVDY 4XBB,*kUq!VBV#B,BM4GYBX X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d mX8kSHyQV0n*Qs,B,/ XB,M,YC[aR>Zle mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G endobj mB&Juib5 <> +9s,BG} <> KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb The sum of the smallest and the . Suppose x and y are odd integers. 'bu +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe log(x+2)log(x1)=log(x+2)log(x1)\frac { \log ( x + 2 ) } { \log ( x - 1 ) } = \log ( x + 2 ) - \log ( x - 1 ) +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk mrJyQ1_ #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ mB,B,R@cB,B,B,H,[+T\G_!bU9VEyQs,B1+9b!C,Y*GVXB[!b!b-,Ne+B,B,B,^^Aub! Consider some even numbers, say, 68, 102. <> *.R_%VWe The smaller of two consecutive integers is eight less than A straightforward word problem solved using an equation. mB&Juib5 WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d *.)ZYG_5Vs,B,z |deJ4)N9 moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l K:QVX,[!b!bMKq!Vl _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b kLq!V KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! Inductive reasoning sequence example, Mouli Javia - StudySmarter Originals.