document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203186>Question 1203186</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 6x7^n - 2x3^n habis dibagi 4 untuk setiap n bilangan asli. \r<BR>\n" );
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document.write( "keterangan : ^n = pangkat n </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #838553 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=838553\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Translation of what the student wrote<BR>\n" );
document.write( "<font color=blue>6x7^n — 2x3^n is divisible by 4 for every n natural numbers.<BR>\n" );
document.write( "description: ^n = power n</font>\r<BR>\n" );
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document.write( "I'll write the solution in English, and let the student translate back to their original native language.\r<BR>\n" );
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document.write( "I'm assuming that you want to prove this claim by induction.\r<BR>\n" );
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document.write( "The natural numbers is the set {1,2,3,4,...}<BR>\n" );
document.write( "n = 1 is the smallest item<BR>\n" );
document.write( "Compute the expression for n = 1<BR>\n" );
document.write( "6*7^n - 2*3^n<BR>\n" );
document.write( "6*7^1 - 2*3^1<BR>\n" );
document.write( "6*7 - 2*3<BR>\n" );
document.write( "42 - 6<BR>\n" );
document.write( "36<BR>\n" );
document.write( "4*9<BR>\n" );
document.write( "The last step shows that we get a multiple of 4<BR>\n" );
document.write( "Which in turn proves that 6*7^n - 2*3^n is a multiple of 4 when n = 1<BR>\n" );
document.write( "In other words, 6*7^n - 2*3^n is divisible by 4 when n = 1\r<BR>\n" );
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document.write( "That covers the base case.<BR>\n" );
document.write( "Now onto the inductive step.<BR>\n" );
document.write( "Assume that 6*7^k - 2*3^k is a multiple of 4, when k is some natural number and k > 1.\r<BR>\n" );
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document.write( "This would mean <BR>\n" );
document.write( "6*7^k - 2*3^k = 4m<BR>\n" );
document.write( "for some integer m\r<BR>\n" );
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document.write( "Based on the assumption above, we want to prove that  6*7^(k+1) - 2*3^(k+1) is also a multiple of 4<BR>\n" );
document.write( "The goal is to show that 6*7^(k+1) - 2*3^(k+1) can be written in the template 4*(some integer).\r<BR>\n" );
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document.write( "Through a bit of algebra...<BR>\n" );
document.write( "6*7^(k+1) - 2*3^(k+1)<BR>\n" );
document.write( "6*7^k*7^1 - 2*3^k*3^1<BR>\n" );
document.write( "42*7^k - 6*3^k<BR>\n" );
document.write( "(6*7^k + 36*7^k) - (2*3^k + 4*3^k)<BR>\n" );
document.write( "6*7^k + 36*7^k - 2*3^k - 4*3^k<BR>\n" );
document.write( "(6*7^k - 2*3^k) + (36*7^k - 4*3^k)<BR>\n" );
document.write( "(4m) + 4*(9*7^k - 1*3^k)<BR>\n" );
document.write( "4( m + 9*7^k - 1*3^k )<BR>\n" );
document.write( "4*(some integer)\r<BR>\n" );
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document.write( "We conclude that 6*7^(k+1) - 2*3^(k+1) is a multiple of 4 when 6*7^k - 2*3^k is a multiple of 4.<BR>\n" );
document.write( "This wraps up the inductive step, and it wraps up the induction proof overall.\r<BR>\n" );
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document.write( "Another approach to the inductive step\r<BR>\n" );
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document.write( "Return back to the assumption of <BR>\n" );
document.write( "6*7^k - 2*3^k = 4m\r<BR>\n" );
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document.write( "Isolating the first term gets us<BR>\n" );
document.write( "6*7^k = 4m + 2*3^k\r<BR>\n" );
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document.write( "Then apply a substitution as marked in <font color=red>red</font><BR>\n" );
document.write( "6*7^(k+1) - 2*3^(k+1)<BR>\n" );
document.write( "6*7^k*7^1 - 2*3^k*3^1<BR>\n" );
document.write( "7*6*7^k - 6*3^k<BR>\n" );
document.write( "7*<font color=red>6*7^k</font> - 6*3^k<BR>\n" );
document.write( "7*(<font color=red>4m + 2*3^k</font>) - 6*3^k<BR>\n" );
document.write( "28m + 14*3^k - 6*3^k<BR>\n" );
document.write( "28m + 8*3^k<BR>\n" );
document.write( "4*( 7m + 2*3^k )<BR>\n" );
document.write( "4*(some integer)\r<BR>\n" );
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document.write( "We have shown that 6*7^(k+1) - 2*3^(k+1) is a multiple of 4 when 6*7^k - 2*3^k is a multiple of 4.\r<BR>\n" );
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document.write( "However you approach the inductive step, both methods show that 6*7^k - 2*3^k being a multiple of 4 leads 6*7^(k+1) - 2*3^(k+1) to be a multiple of 4.\r<BR>\n" );
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document.write( "It is then a domino effect to cause 6*7^n - 2*3^n to be a multiple of 4 for any natural number n.<BR>\n" );
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