e9rX%V\VS^A XB,M,Y>JmJGle Truth value: false; 0 kaqXb!b!BN B,B, *.F* <> K:'G B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX <> The sum of 5 consecutive integers can be 100. _QAXX5l#22!b!b *9B,B,T@seeXU[b)UN,WBW mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab m ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl #Z: You use inductive reasoning when you find a pattern in specific cases and then write a conjecture for the general case. WX+hl*+h:,XkaiC? 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! '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 K,C!+},C!k6YHu!k(^b!b!b=++LtVe&WWX]bY\eYe2dE&X!_!b!b! Using the formula to calculate, the third odd integer is 85, so its 5 times is 5 * 85= 425. StudySmarter is commited to creating, free, high quality explainations, opening education to all. 36 0 obj 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b #Z: 0000055055 00000 n These three situations are discussed separately below. =*GVDY 4XB*VX,B,B,jb|XXXK+ho _)9r_ How do I find the angles of an isosceles triangle whose two base angles are equal and whose third angle is 10 less than three times a base angle? e+D,B1 X:+B,B,bE+ho|XU,[s B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX endstream mrftWk|d/N9 3. #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b BNxmMY ,XF++[aXc!VS _Y}XTY>"/N9"0beU@,[!b!b)N b!VUX)We P(k + 1) is true for all positive integers k. To complete the inductive step, assuming the inductive hypothesis that P(k) holds for an arbitrary integer k, show that must P(k + 1) be true. x+*00P AC(#9KP%+ Let the consecutive numbers be n and n + 1. XH&P|e2d2d^@{WXAb+B,B5 JYY~ +|>kRujJeO,C!+R@{WX&}XXB,,J}>E}W"__aX~'bMj WV]Pi_Ye2dEh 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b <> 4GYc}Wl*9b!U *.N jb!VobUv_!V4&)Vh+P*)B,B!b! >+B,b!pe?dV)+ *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk True. Suppose the sum of four consecutive odd integers is 184. |d/N9 endobj kLqU |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s 0000056879 00000 n m%e+,RVX,B,B)B,B,B LbuU0+B"b s 4XB,,Y mrftWk|d/N9 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. *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 endobj b 4IY?le 'b *.R_ 39 0 obj 16060 OyQ9VE}XGe+V(9s,B,Z9_!b!bjT@se+#}WYlBB,jbM"KqRVXA_!e 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 . mB&Juib5 vkYe&+(Oo~>+(\@kWX5TY,C b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B A:,[(9bXUSbUs,XXSh|d ,[s We _)9r_ C. Prove using deductive reasoning the following conjectures. Given a number N, write a function to express N as sum of two or more consecutive positive numbers. mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b 0000057079 00000 n |d/N9 *. s 4Xc!b!F*b!TY>" kByQ9VEyUq!|+E,XX54KkYqU *. +MrbVkB,B_fiGkeq!V+(F,C,C 49 0 obj ,[s ^[aQX e 16060 kLqU 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: 6++[!b!VGlA_!b!Vl 16060 9b!b=X'b +9Vc}Xq- :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e We Answer (1 of 11): Let the smallest number of the five numbers be x. Nala is an orange cat and she purrs loudly. >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ ?l mrk'b9B,JGC. KJkeqM=X+[!b!b *N ZY@b!b! Let the first number be n #n+(n+1)=5# simplified to #2n=4# divide by 2 gives #n=2 and (n+1)=3# Answer link . nb!Vwb k~u!R_ApV" So, the next dove which comes will also be white. KJkeqM=X+[!b!b *N ZY@b!b! Show a counterexample for the given case to prove its conjecture false. 9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe 33 :X]e+(9sBb!TYTWT\@c)G 4&)kG0,[ T^ZS XX-C,B%B,B,BN *. K:QVX,[!b!bMKq!Vl #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, The product of two consecutive positive integers is 1,332. e9rX%V\VS^A XB,M,Y>JmJGle ,[s 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b Thus, answer choice C A+25 is correct. 'Db}WXX8kiyWX"Qe kLq!V Sign up to highlight and take notes. \text{Then their sum is $5n = 105$. q!VkMy [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e The case which shows the conjecture is false is called a counterexample for that conjecture. How might one go about proving this poorly worded theorem about divisibility with the number 3? which shows that n is sum of ve consecutive integers. stream +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. ,XF++[aXc!VS _Y}XTY>"/N9"0beU@,[!b!b)N b!VUX)We e 'b,N Z=_=Xy!!!b!BbmwyN $}Xq++aIi B]byiK4#_!b!VB X+'+O922B,S@{B !b.O:'Pqyb!V)/MsiOyiJK+B,j^@8ke|b 4XXXXcVvW!B T\^S*.O:'uW_bm-N ZE_!OyiJKKS\?'|XXcV'b|X)O922B,S@B !b.O:'Pqy*9r%t%,)Z@ [5_bn~3;D+dlL._L>; ,S=& endstream endobj 365 0 obj <>stream _,9rkLib!V |d*)M.N B}W:XXKu_!b!b** WP}_o$Te !}XXXGkfY}+(\T+(0Q_A{XHmWSe2dMW!C,BB _!b!b!CV_A X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d SX5X+B,B,0R^Asl2e9rU,XXYb+B,+G 1 + 3 + 5 = 9 1+3+5=9 1 + 3 + 5 = 9. . *. 'bu #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b About Quizlet; How . s 4XB,,Y +GY~W~~1e"!kMu!S;|e2d:~+D XWXXuWX=:Wx +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~~ #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ trailer << /Size 310 /Info 201 0 R /Root 204 0 R /Prev 480387 /ID[<3cd60dd519d6ab5c5d219b7bb7f06c6b><49b31575350be81b6df4257ea55a4316>] >> startxref 0 %%EOF 204 0 obj << /Type /Catalog /Pages 200 0 R /Metadata 202 0 R /AcroForm 206 0 R /Outlines 196 0 R /Names 207 0 R /OpenAction 205 0 R /ViewerPreferences << /HideToolbar true /HideMenubar true >> >> endobj 205 0 obj << /S /GoTo /D [ 210 0 R /FitH -32768 ] >> endobj 206 0 obj << /Fields [ 221 0 R 222 0 R 223 0 R 224 0 R 225 0 R 226 0 R 227 0 R 231 0 R 232 0 R 199 0 R 97 0 R 99 0 R 101 0 R 103 0 R 105 0 R 107 0 R 109 0 R 111 0 R 113 0 R 116 0 R 118 0 R 120 0 R 122 0 R 124 0 R 126 0 R 128 0 R 130 0 R 132 0 R 135 0 R 137 0 R 139 0 R 141 0 R 143 0 R 145 0 R 147 0 R 149 0 R 151 0 R 154 0 R 156 0 R 158 0 R 160 0 R 162 0 R 164 0 R 166 0 R 168 0 R 170 0 R 178 0 R 180 0 R 182 0 R 184 0 R 186 0 R 188 0 R 190 0 R 192 0 R 194 0 R 198 0 R ] /DR << /Font << /ZaDb 197 0 R /Helv 229 0 R >> /Encoding << /PDFDocEncoding 230 0 R >> >> /DA (/Helv 0 Tf 0 g ) >> endobj 207 0 obj << /JavaScript 208 0 R >> endobj 208 0 obj << /Names [ (disclosed)209 0 R ] >> endobj 209 0 obj << /S /JavaScript /JS (this.disclosed = true;\r\n) >> endobj 308 0 obj << /S 957 /O 1549 /V 1565 /Filter /FlateDecode /Length 309 0 R >> stream MX}XX B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ From the above, we can observe that the answer of all the sums is always an even number. mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G *.R_%VWe endstream #Z: I also have seen white geese there. 8Vh+,)MBVXX;V'PCbVJyUyWPq}e+We9B,B1 T9_!b!VX>l% T^ZS X! _ Sum of five consecutive numbers equals . Conjecture: The sum of even numbers is an even number. *.R_%VWe 11 31 3 51 3 5 7 1 12 4 22 9 32 16 42 ANSWER The sum of the first n . 4&)kG0,[ T^ZS XX-C,B%B,B,BN kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! _)9r_ Here, the statements are true, but the conjecture made from it is false. b"b!VW?s|J8J8WXXX+:XB*eeXXM|J8kW5XiJXXO&K|XXX+WWq2B,B,ZY@z+E,C,C GV^Y?le kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b! >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ mX+#B8+ j,[eiXb 62 0 obj !*beXXMBl Here, the conclusion is drawn based on a statistical representation of the sample set. +C,,Hmkk6 XloU'bM +++LWe!!+R@fj*Y2d^@{WX5Xb!b!bMR!0Q_A&j *.)ZYG_5Vs,B,z |deJ4)N9 what connection type is known as "always on"? #rk [a^A 4Xk|do+V@#VQVX!VWBB|X6++B,X]e+(kV+r_ kLqU 6++[!b!VGlA_!b!Vl p}P]U'b!bb9d MxmM=&Pu:VXXkg?W>B XF+4GYkc!b5(O9e+,)M.nj_=#VQ~q!VKb!b:X mrs7+9b!b Rw If so, how close was it? Let n is sum of five consecutive integer of k 2, k-1, k, k + 1, k+2. bbb!b!V_B,B,*.O92Z5k\ WXXX+9r%s%l+C,B,B Xzn *. +9s,BG} N=2d" Yu!>+BB,ZT@uh}2dY_A{WWp}P]U'b} Y endobj kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu XF+4GYkc!b5(O9e+,)M.nj_=#VQ~q!VKb!b:X *.N jb!VobUv_!V4&)Vh+P*)B,B!b! b"b! Therefore,k-2 + k-1 + k + k+1 + k+2= n5*k = nThe five numbers will be n/5 2, n/5 1, n/5, n/5 + 1, n/5 + 2. KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb |d/N9 m%e+,RVX,B,B)B,B,B LbuU0+B"b wl|k^Mx rr,hlX_ *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 _,9rkLib!V |d*)M.N B}W:XXKu_!b!b** *.F* !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!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: _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'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 0000004933 00000 n 34 KW}?*/MI"b!b+j_!b!Vl|*bhl*+]^PrX!XB[aIqDGV4&)Vh+D,B}U+B,XXl*b!Vb The quantity in Column B is greater C. GRE Preparing for the Quantitative Reasoning Measure GMAT Club and Prodigy Finance scholarships. |d/N9 cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X #4GYcm }uZYcU(#B,Ye+'bu Prove that the negative of any even integer is even. b 4IY?le MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G S"b!b A)9:(OR_ 60 + 62 + 64 + 66 + 68 = 320. 6++[!b!VGlA_!b!Vl S: s,B,T\MB,B5$~e 4XB[a_ 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: G 6_!b!V8F)V+9sB6!V4KkAY+B,YC,[o+[ XB,BWX/NQ cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ 15 0 obj Sequence Pattern, Mouli Javia - StudySmarter Originals. *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b *N ZY@AuU^Abu'VWe mB&Juib5 ?*'++a\ B *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* e+D,B1 X:+B,B,bE+ho|XU,[s *Vs,XX$~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 >X@{MxmM]W'|bWse+(VXX[V_!b!b!Te XbbbUn++W5USbB,B,*.OB!lb)UN,WBW .)ZbEe+V(9s,z__WyP]WPqq!s,B,,Y+W+MIZe+(Vh+D,5u]@X2B,ZRBB,Bx=UYo"ET+[a89b!b=XGQ(GBYB[a_ endobj 3W%Xc+^@)B)u.j_bbU'bB,Bty!!!b!}Xb"b!*.Sy8 Step 3 Test your conjecture using other numbers. 59 0 obj mU XB,B% X}XXX++b!VX>|d&PyiM]&PyqlBN!b!B,B,B T_TWT\^Ab endobj endobj Inductive reasoning uses previous examples and patterns to form a conjecture. m5XSYBB,B1!b%+B,GYB[a:_ V,rr&P[}N'CCte ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B *.vq_ ):bKU'bYumkBXO!!k}P]5WcGY~~ endstream N bU+(\TWbe+&+h|N|B,::!!+R@nZ b 18 0 obj cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ We *.N1rV'b5GVDYB[aoiV} T^ZS T^@e+D,B,oQQpVVQs,XXU- MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie 2 is an even number but not composite, as it is a prime number. 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@{ moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l +9Vc}Xq- #BI,WBW ,XF++[aXc!VS _Y}XTY>"/N9"0beU@,[!b!b)N b!VUX)We Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. *.F* |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s GY~~2d}WO !N=2d" XGv*kxu!R_Ap7j(nU__a(>R[SOjY X,CV:nb!b!b! *.*b $$x^3+3x^2+5x+3 =0 \mod 3$$ kaqXb!b!BN 'bu _b!b!b,b_!b!VJ,Cr%$b"b!bm,R_!b!VJSXr%|+B,XX+P\J2 XXXKXXXX s 4XB,,Y #BYB[a+o_@5u]@XB,Bt%VWXX)[aDYXi^}/ kaqXb!b!BN +e+|V+MIB,B,B}T+B,X^YB[aEy/-lAU,X'Sc!buG Are inductive and deductive the same type of reasoning? Find the next 5 terms in the sequence 38, 31, 24, 17, ___, ___, ___, ___, ___ . So, the statements may not always be true in all cases when making the conjecture. WP>+(_X/WeXuLukkY WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d K:QVX,[!b!bMKq!Vl W+,XX58kA=TY>" R22 !!b!b5+/,B,BC,CC 16060 endobj Yes I got it now. |d P,[aDY XB"bC,j^@)+B,BAF+hc=9V+K,Y)_!b P,[al:X7}e+LVXXc:X}XXDb stream b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# Examples: Input : n = 15 Output : 1 2 3 4 5 15 = 1 + 2 + 3 + 4 + 5 Input : n = 18 Output : -1 Recommended: Please try your approach on {IDE} first, before moving on to the solution. So, about 70% of doves are white. +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU _)9r_ +9s,BG} w Click here to see ALL problems on Problems-with-consecutive-odd-even-integers Question 1098921 : If the sum of five consecutive even integers is t, then, in terms of t, what is the greatest integer? 34 mX8@sB,B,S@)WPiA_!bu'VWe cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X _*N9"b!B)+B,BA T_TWT\^AAuULB+ho" X+_9B,,YKK4kj4>+Y/'b kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu e+D,B,ZX@qb+B,B1 LbuU0R^Ab d+We9rX/V"s,X.O TCbWVEBj,Ye 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 |d/N9 <> 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 = 2n . 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 Xg&P|b: X,CV65u]@5zW~XXgV'P>9dF_=a+We&Mx 2duhYHmkk:+G7}WXXufuCV_An,J}Q__a:w@,CV:e&P%'|WXXufuCV_An,J}Q__a:I,CeT'bYWb}+D,B,::AuU_A cEV'PmM UYJK}uX>|d'b b >+[aJYXX&BB,B!V(kV+RH9Vc!b-"~eT+B#8VX_ Show all of your work. 'bu UN=2dd_Y,C!J,BB,Z+B,BU:~+Weu5Y@kWW _!b X!%CVVY,C!J,BB%B,B $TeV+h m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L m endstream 'bub!bC,B5T\TWb!Ve 0dfjWP(0Q_Az&Y!:_Yu!!MxmM]W'bMB,B,R@$AuL_ stream That is mX8@sB,B,S@)WPiA_!bu'VWe ,XF++[aXc!VS _Y}XTY>"/N9"0beU@,[!b!b)N b!VUX)We I appreciate it, We've added a "Necessary cookies only" option to the cookie consent popup. kaqXb!b!BN q!VkMy mrftWk|d/N9 e We have to prove or disprove that the sum of these consecutive integers is divisible by 5 without leaving a remainder. '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 *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD e ~WXUYc9(O j1_9rU,B,58[!_=X'#VX,[tWBB,BV!b=X uWX'VXA,XWe%q_=c+tQs,B58kVX+#+,[BYXUXWXXe+tUQ^AsWBXerkLq! S"b!b A)9:(OR_ W+,XX58kA=TY>" #T\TWT\@W' 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! k^q=X :X B,B, kN}Q__a}5X*0,BBet*eM,C!+R@5)ZFb!b!b=++LtVe&WWX]bY\eYe2dE&XB,B,B9GY~~nPb,B ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl mrk'b9B,JGC. x+*00P A3S0i w K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& Answer by ikleyn(44793) ( Show Source ): Also, the sum of first 'n' positive integers can be calculated as, Sum of first n positive integers = n (n + 1)/2, where n is the total number of integers. #4GYc!,Xe!b!VX>|dPGV{b endobj Ideas: Let n can be written as a, a +1, a +2 .. a + k-1's and (a> = 1), i.e., n = (a + a + k-1) * k / 2. the first term of a gp is twice its common ratio. m b kLq!VH mT\TW X%VW'B6!bC?*/ZGV8Vh+,)N ZY@WX'P}yP]WX"VWe Get the Gauthmath App. 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 endobj <> e mrftWk|d/N9 K:'G ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B 'bub!bC,B5T\TWb!Ve x+*00P A3S0i wI kLq!V OyQ9VE}XGe+V(9s,B,Z9_!b!bjT@se+#}WYlBB,jbM"KqRVXA_!e b9ER_9'b5 'bu q!VkMy ,X'PyiMm+B,+G*/*/N }_ endobj ^[aQX e *. Conjecture Number 20 must be divisible by 5. )+B,:(Vh+LWP&VW|k^MxmM]7WYYzu!pbqXXGU'bM The sum of 5 consecutive integers is equal to 5 times the third integer. SR^AsT'b&PyiM]'uWl:XXK;WX:X mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie 7|d*iGle #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b ~+t)9B,BtWkRq!VXR@b}W>lE #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e 4&)kG0,[ T^ZS XX-C,B%B,B,BN endobj e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e 9b!b=X'b What sort of strategies would a medieval military use against a fantasy giant? 5 0 obj 9b!b=X'b ,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 *.vq_ SR^AsT'b&PyiM]'uWl:XXK;WX:X #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 :e+We9+)kV+,XXW_9B,EQ~q!|d endobj :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! 0000053452 00000 n :X]e+(9sBb!TYTWT\@c)G ,B&PC2d(zu!!++B,::kRJ}+l)0Q_A{WXCVW,Ce^N=2d"b}XXT'bMUp}P]5W~-e&+h *.9r%_5Vs+K,Y>JJJ,Y?*W~q!VcB,B,B,BT\G_!b!VeT\^As9b5"g|XY"rXXc#~iW]#GVwe Solution. Example: Prove the sum of two odd numbers is an even number. 0000003372 00000 n A conjecture is said to be true if it is true for all the cases and observations. X~~ b"V:e^eY,Ce"b!VWXXO$! RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)B* Consider the true statements Numbers ending with 0 and 5 are divisible by 5. +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU kLqU *. JXX+6Jk VX>+kG0oGV4KhlXX{WXX)M|XUV@ce+tUA,XXY_}yyUq!b!Vz~d5Um#+S@e+"b!V>o_@QXVb!be+V9s,+Q5XM#+[9_=X>2 4IYB[a+o_@QXB,B,,[s 0000068151 00000 n *. Create flashcards in notes completely automatically. stream cXB,BtX}XX+B,[X^)R_ ~+t)9B,BtWkRq!VXR@b}W>lE q++aIi *b!VBN!b/MsiU"2B,BA X+WXhg_"b!*.SyU_bm-R_!b/N b!:Oyq\U++C,B,T@B,j_@seeX5&r% +!b!b)O:'Pq}Xkk}X8SXKS\?Ubbb!b!Bb!VC,C,C,B1+a\ kNy'bl'bbb!b\ +JXXsN Tr_!b/9r%t%,)r_!b/N b!:Oy}uXXXX8ke}XkL|JXA,WBB,S@5u*O 'bu The five consecutive integers are 15,16,17,18,19 Explanation: To identify five consecutive integers we begin by giving them each a variable expression 1st = x 2nd = x + 1 3rd = x + 2 4th = x + 3 5th = x + 4 Now we set these equal to a sum of 85 x + x + 1 + x + 2 +x +3 +x +4 = 85 5x +10 = 85 5x+10 10 = 85 10 5x = 75 5x 5 = 75 5 x = 15 Then sketch Do}XXXXKJ,Ckaq=X?b!b!Vqy!!!b$_$++a\ kNyWXX3W%Xo *. e !*beXXMBl 'bub!bC,B5T\TWb!Ve 'b wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U So if any one of the cases is false, the conjecture is considered false. ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl q!Vl "T\TWbe+VWe9rXU+XXh|d*)M|de+'bu #BYB[a+o_@5u]@XB,Bt%VWXX)[aDYXi^}/ 'bub!b)N 0R^AAuUO_!VJYBX4GYG9_9B,ZU@s#VXR@5UJ"VXX: 2eYN5+D,jeT' *C $Pe+k b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX sum of five consecutive integers inductive reasoning gemini and scorpio parents gabi wilson net worth 2021 . *.N jb!VobUv_!V4&)Vh+P*)B,B!b! Consider 2 and 5. 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, k^q=X bbb!b!)z~a!b!b'bbb|X}uXr%D,B9]b!b!bu)9r%t%,iAXXi_=XXX22B,BUSbB,B,*.O922 Step 1: Find a rule by using few examples. x mq]wEuIID\\EwL|4A|^qf9r__/Or?S??QwB,KJK4Kk8F4~8*Wb!b!b+nAB,Bxq! e #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe Therefore, 153 is a neat number. XW+b!5u]@K 4X>l% T^\Syq!Bb!b ** ,X'PyiMm+B,+G*/*/N }_ 'bub!bC,B5T\TWb!Ve 10 0 obj mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs cEZ:Ps,XX$~eb!V{bUR@se+D/M\S stream mX+#B8+ j,[eiXb #4GYc!,Xe!b!VX>|dPGV{b *|eeU+C,B,zb!b!Vqy!!!}_!+a\ ] +JXXS|XXX+g\ ] K|eXX8SbbUWXXH_5%V/,B,BC,C,CB,W"bV 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: 9 0 obj So, the given conjecture is false. + Inductive reasoning consists of the following steps: Observe the sample set and identify the patterns. moIZXXVb5'*VQ9VW_^^AAuU^A 4XoB 4IY>l mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U s 4XB,,Y XXXfq+)ZbEeeUA,C,C,LiJK&kcy_ki5XiJX_!b!VVP+_C_u%!VXXX _fJg\ 6P+^Ob)UN,WBW Describe how to sketch the fourth figure in the pattern. 3 0 obj 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 GV^Y?le 2 The product of three consecutive natural numbers can be equal to their sum. Inductive Reasoning - PDFs. KVX!VB,B5$VWe #Z:(9b!`bWPqq!Vk8*GVDY 4XW|#kG TYvW"B,B,BWebVQ9Vc9BIcGCSj,[aDYBB,ZF;B!b!b!b}(kEQVX,X59c!b!b'b}MY/ #XB[alXMl;B,B,B,z.*kE5X]e+(kV+R@sa_=c+hc!b! e_@s|X;jHTlBBql;B,B,B,Bc:+Zb!Vkb ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl 0000072355 00000 n Solution for 7.2B 1. Which of the above statements is/are correct ? kLqU cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ Connect and share knowledge within a single location that is structured and easy to search. 35 B. mB&Juib5 *Vh+ sWV'3#kC#yiui&PyqM!|e 4XBB,S@B!b5/NgV8b!V*/*/M.NG(+N9 =*GVDY 4XB*VX,B,B,jb|XXXK+ho b 32 0 obj yqUJgV'bmb!V*eeXO$VZJ,Ir%D,B,X@sbXXiJXXq&!b!b!b!g^}%k3WXXX+6 <> There are 10 consecutive nonzero positive integers. Step 3: Test the conjecture for a particular set. We&+(\]SufmMe[}5X+N=2d" W'b_!b!B,CjY}+h *.L*VXD,XWe9B,ZCY}XXC,Y*/5zWB[alX58kD Pattern: Conjecture: _____ Test: DISPROVING CONJECTURES Example 5 Show that the conjecture is false by finding a counterexample. 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 |dEe+_@)bE}#kG TYOkEXXX_)7+++0,[s #rk [a^A 4Xk|do+V@#VQVX!VWBB|X6++B,X]e+(kV+r_ 4GYc}Wl*9b!U $$(3k - 1)((3k - 1)^2+5)=(3k - 1)(9k^2-6k+6)=0 \mod 3$$. .) k~u!AuU_Abe+|(Vh+LT'b6'b9d9dEj(^[S x_9de+|:kRXuH *. ~WXUYc9(O j1_9rU,B,58[!_=X'#VX,[tWBB,BV!b=X uWX'VXA,XWe%q_=c+tQs,B58kVX+#+,[BYXUXWXXe+tUQ^AsWBXerkLq! X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d mrs7+9b!b Rw B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX x+*00P A3S0i wd 0000006092 00000 n k KJs,[aDYBB,R@B,B,B.R^AAuU^AUSbUVXQ^AstWXXe+,)M.Nnq_U0,[BN!b! <> ,BDu! >_=XNu!!MxmM]W'bu+YYmJ!BI!b%CV_An,J}Q__a:w@,CV:e&PX+BB,B3(_T <> WSB3WXXX+WX+B,C,Cr%$b"b!bm,R_!b!VJSXr%D/ Lets once again take a look at what we learned through examples. +hc9(N ZY@s,B,,YKK8FOG8VXXc=:+B,B,ZX@AuU^ATA_!bWe +9Vc}Xq- m"b!bb!b!b!uTYy[aVh+ sWXrRs,B58V8i+,,Ye+V(L *.R_%VWe Formula for sum of 'n' terms of an arithmetic sequence: S n = n 2 [ 2 a 1 + ( n - 1) d]. 34 VX>+kG0oGV4KhlXX{WXX)M|XUV@ce+tUA,XXY_}yyUq!b!Vz~d5Um#+S@e+"b!V>o_@QXVb!be+V9s,+Q5XM#+[9_=X>2 4IYB[a+o_@QXB,B,,[s #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ d+We9rX/V"s,X.O TCbWVEBj,Ye *.)ZYG_5Vs,B,z |deJ4)N9 S: s,B,T\MB,B5$~e 4XB[a_ #TA_!b)Vh+(9rX)b}Wc!bM*N9e+,)MG"b 5(n +2) If we divide this sum of any 5 consecutive integers by 5 we get: 5(n + 2) 5 . b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B XW+b!5u]@K 4X>l% T^\Syq!Bb!b ** cXB,BtX}XX+B,[X^)R_ "T\TWbe+VWe9rXU+XXh|d*)M|de+'bu If you preorder a special airline meal (e.g. '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 b>X+B,XX+P\D2 b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# x -qo@"EyCv?Oc?/?='rvx`??j; . *. k 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 . K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& #4GYc!,Xe!b!VX>|dPGV{b 3. :e+We9+)kV+,XXW_9B,EQ~q!|d RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)B* m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU _,9rkLib!V |d*)M.N B}W:XXKu_!b!b** ,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 _,9rkLib!V |d*)M.N B}W:XXKu_!b!b** Step 3 Test your conjecture using other numbers. 6++[!b!VGlA_!b!Vl +9s,BG} I need to deductively prove that the sum of cubes of $3$ consecutive natural numbers is divisible by $9$. WX+hl*+h:,XkaiC? e A:,[(9bXUSbUs,XXSh|d 'bu !MU'b :X]e+(9sBb!TYTWT\@c)G |d/N9 0000152179 00000 n m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> Free and expert-verified textbook solutions. k~u!l e+D,B1 X:+B,B,bE+ho|XU,[s 'bu State the smaller odd integer x. Remember: Consecutive numbers are numbers that come after another in increasing order. X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d b9ER_9'b5 _)9r_ What is the symbolic form of and inverse statement? w *.R_ e :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e 'Db}WXX8kiyWX"Qe >> :X]e+(9sBb!TYTWT\@c)G W+,XX58kA=TY>" b"bu#VCXXX/-9r%_b!b!b,N T B| }XXbbb!b#VBJXXJ+ZXiJXX&bu !VJ|eXX8S Xj2k~$b"b!bm,O92z+MrbV+E_ What is the third number? mrftWk|d/N9 ?l ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl 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 wV__a(>R[S3}e2dN=2d" XGvB,ZW@5)WP>+(J[WW=++D!zYHu!!N :|5WYX&X b9rXKyP]WPqq!Vk8*GVDYmXiMRVX,B,Lkni V+bEZ+B ^[aQX e Selection type: the default is consecutive integers, of course, you can choose even or odd numbers according to your needs. Let us first identify the observation and hypothesis for this case. 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: 34 #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= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d .)ZbEe+V(9s,z__WyP]WPqq!s,B,,Y+W+MIZe+(Vh+D,5u]@X2B,ZRBB,Bx=UYo"ET+[a89b!b=XGQ(GBYB[a_ :X]e+(9sBb!TYTWT\@c)G q!VkMy K:'G 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 0000172261 00000 n #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 e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d m% XB,:+[!b!VG}[ cB ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e e+D,B,ZX@qb+B,B1 LbuU0R^Ab #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, e %PDF-1.4 stream S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! K|,[aDYB[!b!b B,B,B 4JYB[y_!XB[acR@& Ne^@2dY]S9_=BYu!U}WW _; s 4XB,,Y two separate circles that show that the two items have no relation, phil 305 midterm: kant, utilitarian, locke, s. s 4XB,,Y #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 Uu!zu@,C!UMxmM=tj(^]S$_]zBI!b!1 ?*'++a\ nsB,B,BN!VWO:XX_!bXXXX#|JJAC/ m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> + #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ B,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb &Pk(^@ud|Vu!BC+B2lWP>+(\_ANe+(\_A{;b!1rZ_[S=d&P:!VMxuM!5X+Zb!B#(_TWF_! $Te #AU+JVh+ sW+hc!b52 4XB[aIqVUGVJYB[alX5}XX B,B%r_!bMPVXQ^AsWRrX.O9e+,i|djO,[8S bWX B,B+WX"VWe 7|d*iGle 65 0 obj B,B= XBHyU=}XXW+hc9B]:I,X+]@4Kk#klhlX#}XX{:XUQTWb!Vwb X8keqUywW5,[aVvW+]@5#kgiM]&Py|e 4XB[aIq!Bbyq!z&o?A_!+B,[+T\TWT\^A58bWX+hc!b!5u]BBh|d #4GYc!bM)R_9B 4X>|d&PyiM]&PyqSUGVZS/N b!b-)j_!b/N b!VEyP]WPqy\ 0000070192 00000 n e9rX |9b!(bUR@s#XB[!b!BNb!b!bu KbRVX,X* VI-)GC,[abHY?le knXX5L S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu 0000053987 00000 n &4XS5s*,BDW@kWX5TY,CN!V@uWXQb!b=X_+B,@bMU! X+WBW 0000174791 00000 n =B,BEb!N= cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ Now we test this conjecture on another sequence to consider if the derived conclusion is in fact true for all consecutive numbers.
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