document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1204431>Question 1204431</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine whether 280 is a term in the G.P 1,3,9,27… </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #840674 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=840674\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Method 1\r<BR>\n" );
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document.write( "first term = 1<BR>\n" );
document.write( "common ratio = 3 (since 3/1 = 3, 9/3 = 3, and 27/9 = 3, etc)\r<BR>\n" );
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document.write( "Triple each term to get the next term. Extend the sequence until reaching 280 or going past it.\r<BR>\n" );
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document.write( "1, 3, 9, 27, 81, 243, 729, ...\r<BR>\n" );
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document.write( "We do not reach 280.\r<BR>\n" );
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document.write( "Tip: Use spreadsheet software to generate that list quickly. <BR>\n" );
document.write( "The <font color=red>sequence</font> command in GeoGebra is another route using technology. There are many online tools that can be used as well.\r<BR>\n" );
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document.write( "Method 2\r<BR>\n" );
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document.write( "The nth term of a geometric progression (GP) is: a*(r)^(n-1)<BR>\n" );
document.write( "a = first term<BR>\n" );
document.write( "r = common ratio\r<BR>\n" );
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document.write( "Because a = 1 and r = 3, we get 1*(3)^(n-1) or simply 3^(n-1)\r<BR>\n" );
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document.write( "Set this equal to 280 and solve for n.<BR>\n" );
document.write( "Because the exponent is in the trees, we'll need to log it down.\r<BR>\n" );
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document.write( "3^(n-1) = 280<BR>\n" );
document.write( "log( 3^(n-1) ) = log(280)<BR>\n" );
document.write( "(n-1)*log(3) = log(280)<BR>\n" );
document.write( "n-1 = log(280)/log(3)<BR>\n" );
document.write( "n-1 = 5.1290065<BR>\n" );
document.write( "n = 5.1290065+1<BR>\n" );
document.write( "n = 6.1290065\r<BR>\n" );
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document.write( "We do not achieve a whole number solution for n, so it is impossible to reach 280 when plugging in whole numbers to 3^(n-1)\r<BR>\n" );
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document.write( "Method 3\r<BR>\n" );
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document.write( "Let's assume 280 is a term somewhere in 1,3,9,27,...\r<BR>\n" );
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document.write( "If so, then repeatedly dividing 280 over powers of 3 should get us to the terms your teacher gave you.\r<BR>\n" );
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document.write( "280/3 = 93.3333<BR>\n" );
document.write( "We run into a problem almost immediately.<BR>\n" );
document.write( "The non-whole number result would mean that repeated \"divide by 3\" operations would further push the result into non-whole number territory.\r<BR>\n" );
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document.write( "Sure enough,<BR>\n" );
document.write( "93.3333/3 = 31.1111<BR>\n" );
document.write( "31.1111/3 = 10.3703667<BR>\n" );
document.write( "10.3703667/3 = 3.4567889<BR>\n" );
document.write( "3.4567889/3 = 1.152262967\r<BR>\n" );
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document.write( "The results are somewhat close to 1,3,9,27,... but not an exact match.\r<BR>\n" );
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document.write( "This proves that 280 is not part of the geometric sequence 1,3,9,27,...\r<BR>\n" );
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document.write( "Method 4\r<BR>\n" );
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document.write( "Perhaps the easiest method of the list. No calculators are needed. No need for helpful tech to extend a list of values.\r<BR>\n" );
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document.write( "Notice that the subsequence 3,9,27,... are multiples of 3. So we can use the divisibility by 3 rule.\r<BR>\n" );
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document.write( "Add the digits of 280<BR>\n" );
document.write( "2+8+0 = 10<BR>\n" );
document.write( "Repeat the process of adding the digits<BR>\n" );
document.write( "1+0 = 1<BR>\n" );
document.write( "The digit sum of 1 is not a multiple of 3, so we know that 280 is not a multiple of 3.\r<BR>\n" );
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document.write( "Therefore, it is impossible to reach 280 when writing powers of 3.\r<BR>\n" );
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document.write( "Conclusion: 280 is NOT a term of the GP 1,3,9,27,...<BR>\n" );
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