x 1 (x 1 1) 1 (x 1 2) 1 (x 1 3) 1 (x 1 4) 5 __?__ 22. Make and test conjecture for the sum of two even numbers. <> *.N1rV'b5GVDYB[aoiV} T^ZS T^@e+D,B,oQQpVVQs,XXU- |d/N9 let a and b be odd numbers. <> cEV'bUce9B,B'*+M.M*GV8VXXch>+B,B,S@$p~}X Let's take a look at some of the advantages and limitations of inductive reasoning. mX8@sB,B,S@)WPiA_!bu'VWe )_a:kY5!V@e+L(++B,7XS5s*,BD}VE}WN5+D,C!kxuY}e&&e KVX!VB,B5$VWe 6++[!b!VGlA_!b!Vl b"b!.
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. :X]e+(9sBb!TYTWT\@c)G 0000151388 00000 n
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9Vc!b-"e}WX&,Y% 4XB*VX,[!b!b!V++B,B,ZZ^Ase+tuWO Qe The sum of 5 consecutive integers can be 100. b 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>+(\@$!u_! Yk(^[S3k 'bul"b U}#*+[aXw+h|B,:XY}XuC,,[a65XsWT'bY]Si_!bNU How to Sum Integers 1 to n. You dont need to be a math whiz to be a good programmer, but there are a handful of equations you will want to add to your problem solving toolbox. s 4XB,,Y :X 6++[!b!VGlA_!b!Vl In math, what does reciprocal mean? bWb!b!b!b!bWO V^S*.12B,B,}JXX+"22'+Msi$b"b!b5B,B,z&*'++ay%0B,B,B,B,z@N T\?c|eXX/j5UWbbEeeuWO VR)/Ir%D,B,j}XXLb)UN,WBW X2dU+(\TW__aX~We"V65oW,C!^@{e+D,Z+B,W'bMUp}P]Fb&WN}Q_!bEj(^[S;o{e2d X,BBBI*_aKY~~ B&R^As+A,[Xc!VSFb!bVlhlo%VZPoUVX,B,B,jSbXXX cB C. Prove using deductive reasoning the following conjectures. kByQ9V8ke}uZYc!b=X&PyiM]&Py}#GVC,[!b!bi'bu endstream No need to think about the whole process. We ++m:I,X'b &PyiM]g|dhlB X|XXkIqU=}X buU0R^AAuU^A X}|+U^AsXX))Y;KkBXq!VXR@8lXB,B% LbEB,BxHyUyWPqqM =_ 'bul"b Which of the above statements is/are correct ? m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> b9ER_9'b5 kLq!VH ,B,HiMYZSbhlB XiVU)VXXSV'30
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UYJK}uX>|d'b 14 0 obj |d/N9 where a 1 - first term d is the common difference Types of Consecutive Integers Depending upon the type of integer, the different types of consecutive integers are as follows: Odd Consecutive Integers Even Consecutive Integers Positive Consecutive Integers [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 If yes, find the five consecutive integers, else print "-1". *. *.N jb!VobUv_!V4&)Vh+P*)B,B!b! So if any one of the cases is false, the conjecture is considered false. ++cR@&B_!b'~e 4XB[aIq!+[HYXXS&B,Bxq!Vl 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b mX+#B8+ j,[eiXb XW+b!5u]@K 4X>l% T^\Syq!Bb!b
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Y mrAU+XBF!pb5UlW>b 4IYB[aJ}XX+bEWXe+V9s A:,[(9bXUSbUs,XXSh|d XF+4GYkc!b5(O9e+,)M.nj_=#VQ~q!VKb!b:X 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 m% XB,:+[!b!VG}[ cB !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!VWe mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, #Z: e+D,B,ZX@qb+B,B1 LbuU0R^Ab A:,[(9bXUSbUs,XXSh|d |d/N9 *.)ZYG_5Vs,B,z |deJ4)N9 6XXX *.J8j+hc9B,S@5,BbUR@5u]@X:XXKVWX5+We9rX58KkG'}XB,YKK8ke|e 4XBB,S@B!b5/N* +9Vc}Xq- +e+|V+MIB,B,B}T+B,X^YB[aEy/-lAU,X'Sc!buG .)ZbEe+V(9s,z__WyP]WPqq!s,B,,Y+W+MIZe+(Vh+D,5u]@X2B,ZRBB,Bx=UYo"ET+[a89b!b=XGQ(GBYB[a_ endstream 34 mX8kSHyQV0n*Qs,B,/ XB,M,YC[aR>Zle MX[_!b!b!JbuU0R^AeC_=XB[acR^AsXX)ChlZOK_u%Ie
mB,B,R@cB,B,B,H,[+T\G_!bU9VEyQs,B1+9b!C,Y*GVXB[!b!b-,Ne+B,B,B,^^Aub! 4GYc}Wl*9b!U kLqU e b9B,J'bT/'b!b!*GVZS/N)M,['kEXX# 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 cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ 'bu 34 x mq]wEuIID\\EwL|4A|^qf9r__/Or?S??QwB,KJK4Kk8F4~8*Wb!b!b+nAB,Bxq! endobj \begin{align*} *.N1rV'b5GVDYB[aoiV} T^ZS T^@e+D,B,oQQpVVQs,XXU- Using the formula to calculate, the third odd integer is 85, so its 5 times is 5 * 85= 425. 6'bbb!b0+WBWBB,ZY@5ukOq++aIi V+_!b!BN!b/Ms}eeU+C,B,T@WXW_"b!*.S=}XX{g\ ] KJZ mrftWk|d/N9 SX5X+B,B,0R^Asl2e9rU,XXYb+B,+G 0000005175 00000 n
_TAXXWWeeUA,C,C,B,ZXTs|XX5k9*|XiJXX5J}XX
B@q++aIqYU kaqXb!b!BN #-bhl*+r_})B,B5$VSeJk\YmXiMRVXXZ+B,XXl b *. RR^As9VEq!9bM(O TCbWV@5u]@lhlX5B,_@)B* [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e The positive difference of the cubes of two consecutive positive integers is 111 less than five times the product of the two consecutive integers. ZkwqWXX4GYBXC$VWe9(9s,Bk*|d#~q!+CJk\YBB,B6!b#}XX5(V;+[HYc!b!*+,YhlBz~WB[alXX+B,B1 4JYB[aEywWB[ao" XmB,*+,Yhl@{ Number 20 ends with 0. sum of five consecutive integers inductive reasoning. Thus, answer choice C A+25 is correct. +"b!Bu+B,W'*e 8VX0E,[kLq!VACB,B,B,z4*V8+,[BYcU'bi99b!V>8V8x+Y)b cE+n+-: s,B,T@5u]K_!u8Vh+DJPYBB,B6!b=XiM!b!,[%9VcR@&&PyiM]_!b=X>2 4XB[!bm wJ mB&Juib5 0000151930 00000 n
The difference between two numbers is always less than its sum. A:,[(9bXUSbUs,XXSh|d ^,9Z:WPqqM!G9b!b*M.M*/hlBB1 X}b!bC,B5T\TWAu+B +GYc!b}>_!CV:!VN ::YYmMXX: mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs How might one go about proving this poorly worded theorem about divisibility with the number 3? b"bygXXXW XXXUbYK&kcyXqV!k6*'++a\ endobj 'b !bWVXr_%p~=9b!KqM!GVweFe+v_J4&)VXXB,BxX!VWe *.R_ XW+b!5u]@K 4X>l% T^\Syq!Bb!b
** Prove that a group of even order must have an element of order 2. s 4XB,,Y 'bu q!VkMy Its like a party trick for technical interviews. The general unproven conclusion we reach using inductive reasoning is called a conjecture or hypothesis. <> XbbbUn++W5USbB,B,*.OB!lb)UN,WBW m% XB,:+[!b!VG}[ !*beXXMBl U3}WR__a(+R@2d(zu!__!b=X%_!b!9 LbMU!R_Aj e9rX%V\VS^A XB,M,Y>JmJGle :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e A:,[(9bXUSbUs,XXSh|d SR^AsT'b&PyiM]'uWl:XXK;WX:X e9z9Vhc!b#YeB,*MIZe+(VX/M.N B,jb!b-b!b!(e UXWXXe+VWe
>zl2e9rX5kGVWXW,[aDY X}e+VXXcV 'bk|XWPqyP]WPq}XjHF+kb}X T^ZSJKszC,[kLq! mrJyQb!y_9rXX[hl|dEe+V(VXXB,B,B} Xb!bkHF+hc=XU0be9rX5Gs cB We noticed that, starting with 3, every second number can be written as a sum of two consecutive numbers; starting with 6, every third number (6, 9, 12 . SX5X+B,B,0R^Asl2e9rU,XXYb+B,+G Observation: The product of the two numbers is positive. We :e+WeM:Vh+,S9VDYk+,Y>*e+_@s5c+L&$e 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: kLq!V>+B,BA Lb !*beXXMBl A:,[(9bXUSbUs,XXSh|d cXB,BtX}XX+B,[X^)R_ W+,XX58kA=TY>" [++LWe!!+R@zoeZ,C X~X+B,::I9dp}P]5Ww0A,w+hMxmM!*CVX,CV:@bAXXV'35UY3fNb&WN}Qb" ~Yse2dEh Given that $a$, $b$, $c$ are natural numbers, with $a^2+b^2=c^2$ and $ c-b=1$, prove the following. 15717 ~WXUYc9(O j1_9rU,B,58[!_=X'#VX,[tWBB,BV!b=X
uWX'VXA,XWe%q_=c+tQs,B58kVX+#+,[BYXUXWXXe+tUQ^AsWBXerkLq! XXX22B,E}JJB,O4JJXA,WBBjb}WXX) mB&Juib5 Given an integer n, the task is to find whether n can be expressed as sum of five consecutive integer. ^[aQX e Let us take into consideration the integer numbers. _~WXXX)B,@w Consecutive Integers can be written in the form: n, n + 1, n + 2, etc, where n is an integer. 0000057583 00000 n
*.N jb!VobUv_!V4&)Vh+P*)B,B!b! p}P]WPAuUOQ_ X>+kG0,[!b}X!*!b |X+B,B,,[aZ)=zle9rU,B,%|8g
TY=?*W~q5!{}4&)Vh+D,B} XbqR^AYeE|X+F~+tQs,BJKy'b5 mrk'b9B,JGC. nb!Vwb We wQl8SXJ}X8F)Vh+(*N l)b9zMX%5}X_Yq!VXR@8}e+L)kJq!Rb!Vz&*V)*^*0E,XWe!b!b|X8Vh+,)MB}WlX58keq8U D:U!_;GY_+ZC,B :X We endstream )+B,:(Vh+LWP&VW|k^MxmM]7WYYzu!pbqXXGU'bM 6XjNo|Xq++aIi B]byiK4XOb!bV'b@kLq! m b <> S"b!b A)9:(OR_ VX>+kG0oGV4KhlXX{WXX)M|XUV@ce+tUA,XXY_}yyUq!b!Vz~d5Um#+S@e+"b!V>o_@QXVb!be+V9s,+Q5XM#+[9_=X>2 4IYB[a+o_@QXB,B,,[s mT\TW XuW+R@&BzGV@GVQq!VXR@8F~}VYiM+kJq!k*V)*jMV(G Through the above discussion, you should understand how to calculate the sum of 5 consecutive integers. Then state the truth value 9b!b=X'b <>
sum of five consecutive integers inductive reasoning *.R_ S XA 2, 2 XB} 1 2}, 2 XC 3, 10 XD 2, 21 23. Solution List some examples and look for a pattern. b9zRTWT\@c9b!blEQVX,[aXiM]ui&$e!b!b! 35 +++LtU}h
$$(3k - 1)((3k - 1)^2+5)=(3k - 1)(9k^2-6k+6)=0 \mod 3$$. 20 0 obj #rk [a^A 4Xk|do+V@#VQVX!VWBB|X6++B,X]e+(kV+r_ #4GYcm }uZYcU(#B,Ye+'bu *./)z*V8&_})O jbeJ&PyiM]&Py|#XB[!b!Bb!b
*N ZY@AuU^Abu'VWe b9ER_9'b5 51 0 obj mrk'b9B,JGC. &!t_}W[SSH+D,jXeb-X'bj *,BD[}P]WP:YmbYNw5e:,BBvUg?Y@udm k ~+t)9B,BtWkRq!VXR@b}W>lE +e+D:+[kEXFYB[aEyuVVl+AU,X'P[bU +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk endobj Here the difference between two numbers 2 and 3 is greater than its sum. 4. KJkeqM=X+[!b!b
*N ZY@b!b! *.N jb!VobUv_!V4&)Vh+P*)B,B!b! SR^AsT'b&PyiM]'uWl:XXK;WX:X UyA q++aIi *b!VBN!b/MsiP"2B,BA X+[WXh_"b!*.SyQ_bm-R_!b/N b!:OyqU++C,B,T@}XkLq++!b!b,O:'Pqy By registering you get free access to our website and app (available on desktop AND mobile) which will help you to super-charge your learning process. mB&Juib5 c++D,CCY,CV_YY~5:H_!b!bRC_a(_0,BB2dN=:a*_Y wV__a(>R[S3}e2dN=2d" XGvB,ZW@5)WP>+(J[WW=++D!zYHu!!N :|5WYX&X It's true when $x=0 \mod 3$. #Z:'b f}XGXXk_Yq!VX9_UVe+V(kJG}XXX],[aB, #BYB[a+o_@5u]@XB,Bt%VWXX)[aDYXi^}/ 16060 bbb!b!)z~a!b!b'bbb|X}uXr%D,B9]b!b!bu)9r%t%,iAXXi_=XXX22B,BUSbB,B,*.O922 [as4l*9b!rb!s,B4|d*)N9+M&Y#e+"b)N TXi,!b '(e +X}e+&Pyi V+b|XXXFe+tuWO 0T@c9b!b|k*GVDYB[al}K4&)B,B,BN!VDYB[y_!Vhc9 s,Bk WGe+D,B,ZX@B,_@e+VWPqyP]WPq}uZYBXB6!bB8Vh+,)N Zz_%kaq!5X58SHyUywWMuTYBX4GYG}_!b!h|d Andy and Belen made stars at the same time, making a total of 160 stars in 8 minutes. #4GYcm }uZYcU(#B,Ye+'bu endobj m% XB0>B,BtXX#oB,B,[a-lWe9rUECjJrBYX%,Y%b- YiM+Vx8SQb5U+b!b!VJyQs,X}uZYyP+kV+,XX5FY> MX}XX
B,j,[J}X]e+(kV+R@&BrX8Vh+,)j_Jk\YB[!b!b AXO!VWe S4GYkLiu-}XC,Y*/B,zlXB,B% X|XX+R^AAuU^AT\TW0U^As9b!*/GG}XX>|d&PyiM]'b!|e+'bu kbyUywW@YHyQs,XXS::,B,G*/**GVZS/N b!b-'P}yP]WPq}Xe+XyQs,X X+;:,XX5FY>&PyiM]&Py|WY>"/N9"b!
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