document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=836817>Question 836817</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove by induction that for all n (n being positive natural numbers), \r<BR>\n" );
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document.write( "a) (x^n) - (y^n) is a multiple of (x-y), where x does not equal y, and x and y are integers.\r<BR>\n" );
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document.write( "b) (Sigma n, i=1) i x i! = (n+1)! - 1\r<BR>\n" );
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document.write( "This is a discrete mathematics word problem. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #504376 by </B><A HREF=/tutors/aboutme.mpl?userid=stanbon><B>stanbon(75887)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=stanbon><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=504376\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  Prove by induction that for all n (n being positive natural numbers), <BR>\n" );
document.write( "a) (x^n) - (y^n) is a multiple of (x-y), where x does not equal y, and x and y are integers.<BR>\n" );
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document.write( "1st: Show it is true for n = 1<BR>\n" );
document.write( "(x^1)-(y^1) = 1(x-y):::: true<BR>\n" );
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document.write( "2nd: Assume it is true for n = k::<BR>\n" );
document.write( "(x^k)-(y^k) = m(x-y) where m is an integer<BR>\n" );
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document.write( "3rd: Prove it is true for n = k+1<BR>\n" );
document.write( "(x^(k+1))-(y^(k+1)) = (x^k)*x - (y^k)*y<BR>\n" );
document.write( "= (x^k)-(y^k) + x-y <BR>\n" );
document.write( "Then (x^k-y^k) = m(x-y) and (x-y) = 1*(x-y)<BR>\n" );
document.write( "So, (x^(k+1))-(y^(k+1)) = (m+1(x-y)<BR>\n" );
document.write( "QED<BR>\n" );
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document.write( "b) (Sigma n, i=1) i x i! = (n+1)! - 1<BR>\n" );
document.write( "1st Show for i = 1<BR>\n" );
document.write( "1x1! = (1+1)!-1<BR>\n" );
document.write( "1*1 =  = 2!-1 = 1<BR>\n" );
document.write( "1 = 1<BR>\n" );
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document.write( "2nd: Assume true for i = k<BR>\n" );
document.write( "k*k! = (k+1)! -1<BR>\n" );
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document.write( "3rd: Prove true for i = k+1<BR>\n" );
document.write( "(k+1)(k+1)!<BR>\n" );
document.write( "= k(k+1)! + (k+1)!<BR>\n" );
document.write( "= k(k+1)k! + (k+1)k!<BR>\n" );
document.write( "= (k+1)[k(k!] + k*k!+ k!<BR>\n" );
document.write( "= (k+1+1][k*k!] + k!<BR>\n" );
document.write( "From 2nd you get::<BR>\n" );
document.write( "= (k+2][(k+1)!-1] + k!<BR>\n" );
document.write( "= (k+2)(k+1)! - (k+2) + k!<BR>\n" );
document.write( "= (k+2)! - (k+2)+k!<BR>\n" );
document.write( "Comment:: That does not seem to be working out.<BR>\n" );
document.write( "Please check it out.<BR>\n" );
document.write( "==============<BR>\n" );
document.write( "Cheers,<BR>\n" );
document.write( "Stan H.<BR>\n" );
document.write( "=============================== </SPAN>\n" );
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