document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1155514>Question 1155514</A>: <I><SPAN STYLE=\"word-break: break-all;\"> By the method of induction, show that<BR>\n" );
document.write( "10^n+3.4^(n+1)+5 is divisible by 9 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #778121 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=778121\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'm assuming that decimal point is intended to be a multiplication sign. <BR>\n" );
document.write( "I'm going to use an asterisk to mean multiplication.<BR>\n" );
document.write( "3*4 = 3 times 4\r<BR>\n" );
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document.write( "Base Case: Plug in n = 1 as this is the smallest number in the set of natural numbers {1,2,3,4,...}\r<BR>\n" );
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document.write( "10^n+3*4^(n+1)+5 = 10^1+3*4^(1+1)+5<BR>\n" );
document.write( "10^n+3*4^(n+1)+5 = 10^1+3*4^2+5<BR>\n" );
document.write( "10^n+3*4^(n+1)+5 = 10+3*16+5<BR>\n" );
document.write( "10^n+3*4^(n+1)+5 = 10+48+5<BR>\n" );
document.write( "10^n+3*4^(n+1)+5 = 58+5<BR>\n" );
document.write( "10^n+3*4^(n+1)+5 = 63<BR>\n" );
document.write( "10^n+3*4^(n+1)+5 = 9*7\r<BR>\n" );
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document.write( "We have shown that 10^n+3*4^(n+1)+5 is a multiple of 9 when n = 1.<BR>\n" );
document.write( "So 10^n+3*4^(n+1)+5 is divisible by 9 when n = 1.\r<BR>\n" );
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document.write( "The base case is confirmed.<BR>\n" );
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document.write( "Inductive Step:\r<BR>\n" );
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document.write( "Assume that 10^n+3*4^(n+1)+5 is divisible by 9 for some general case n = k, where k is an integer.\r<BR>\n" );
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document.write( "This basically means, <BR>\n" );
document.write( "10^k+3*4^(k+1)+5 = 9m<BR>\n" );
document.write( "for some integer m\r<BR>\n" );
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document.write( "What we need to prove is that based on that assumption is the n = k+1 case holds true as well. In other words, if we assume P(k) is divisible by 9, then we must show this leads to P(k+1) being divisible by 9 as well. In this case, P(n) = 10^n+3*4^(n+1)+5\r<BR>\n" );
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document.write( "We will use the equation 10^k+3*4^(k+1)+5 = 9m to help show that P(k+1) is divisible by 9. Isolate 10^k to get<BR>\n" );
document.write( "10^k+3*4^(k+1)+5 = 9m<BR>\n" );
document.write( "10^k+5 = 9m-3*4^(k+1)<BR>\n" );
document.write( "10^k = 9m-3*4^(k+1)-5<BR>\n" );
document.write( "This will help with a substitution we'll do later.\r<BR>\n" );
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document.write( "Now let's go back to 10^n+3*4^(n+1)+5 and plug in n = k+1<BR>\n" );
document.write( "The goal is to write the expression in the form 9q, where q is some integer.\r<BR>\n" );
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document.write( "10^n+3*4^(n+1)+5<BR>\n" );
document.write( "10^(k+1)+3*4^(k+1+1)+5 ... every n replaced with k+1<BR>\n" );
document.write( "10^k*10^1+3*4^(k+1)*4^1+5<BR>\n" );
document.write( "10^k*10+3*4^(k+1)*4+5<BR>\n" );
document.write( "10*10^k+3*4*4^(k+1)+5<BR>\n" );
document.write( "10*(10^k)+3*4*4^(k+1)+5<BR>\n" );
document.write( "10*(9m-3*4^(k+1)-5)+3*4*4^(k+1)+5 ... replace 10^k with 9m-3*4^(k+1)-5<BR>\n" );
document.write( "10*(9m)+10(-3*4^(k+1))+10(-5)+3*4*4^(k+1)+5<BR>\n" );
document.write( "90m-30*4^(k+1)-50+12*4^(k+1)+5<BR>\n" );
document.write( "90m + (-50+5) + (-30*4^(k+1)+12*4^(k+1))<BR>\n" );
document.write( "90m - 45 + (-30+12)*4^(k+1)<BR>\n" );
document.write( "90m - 45  - 18*4^(k+1)<BR>\n" );
document.write( "9*10m - 9*5  - 9*2*4^(k+1)<BR>\n" );
document.write( "9*(10m - 5  - 2*4^(k+1))<BR>\n" );
document.write( "9q\r<BR>\n" );
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document.write( "We have shown that if 10^k+3*4^(k+1)+5 = 9m, then 10^(k+1)+3*4^(k+1+1)+5 = 9q where q = 10m - 5  - 2*4^(k+1)<BR>\n" );
document.write( "In other words, we have just shown that if P(k) = 9m, then P(k+1) = 9q, for integers m and q.\r<BR>\n" );
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document.write( "This wraps up the inductive step. Also, it concludes the proof. <BR>\n" );
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