document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1090201>Question 1090201</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Could you please explain the procedure in solving this problem:\r<BR>\n" );
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document.write( "a1=7<BR>\n" );
document.write( "an=224<BR>\n" );
document.write( "Sn=441<BR>\n" );
document.write( "n=? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #704613 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=704613\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\"><BR>\n" );
document.write( "You haven't stated whether this is an arithmetic sequence or a geometric sequence. \r<BR>\n" );
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document.write( "Let's assume for a moment that the sequence is arithmetic. If that's the case, then we'd use the formula<BR>\n" );
document.write( "Sn = (n/2)*(a1+an)\r<BR>\n" );
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document.write( "Plug in the given info and solve for n<BR>\n" );
document.write( "Sn = (n/2)*(a1+an)<BR>\n" );
document.write( "441 = (n/2)*(7+224)<BR>\n" );
document.write( "441 = (n/2)*231<BR>\n" );
document.write( "441/231 = n/2<BR>\n" );
document.write( "n/2 = 441/231<BR>\n" );
document.write( "n/2 = 21/11<BR>\n" );
document.write( "n = 2*(21/11)<BR>\n" );
document.write( "n = 42/11<BR>\n" );
document.write( "n = 3.81818181818181 (approximate)\r<BR>\n" );
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document.write( "The fact that n is NOT a whole number indicates that the sequence is NOT arithmetic. \r<BR>\n" );
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document.write( "Let's assume that the sequence is geometric instead.\r<BR>\n" );
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document.write( "The pattern of geometric terms (a1,a2,a3,...) look like this<BR>\n" );
document.write( "a1 = 7<BR>\n" );
document.write( "a2 = a1*r = 7*r<BR>\n" );
document.write( "a3 = a2*r = (7r)*r = 7r^2 = 7r^(3-1)<BR>\n" );
document.write( "a4 = a3*r = (7r^2)*r = 7r^3 = 7r^(4-1)<BR>\n" );
document.write( "a5 = a4*r = (7r^3)*r = 7r^4 = 7r^(5-1)<BR>\n" );
document.write( "Each term is the result of multiplying the common ratio r by the previous term<BR>\n" );
document.write( "Take note how the exponents are formed.\r<BR>\n" );
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document.write( "The pattern continues forever. The nth term is<BR>\n" );
document.write( "an = a1*r^(n-1)<BR>\n" );
document.write( "an = 7*r^(n-1)\r<BR>\n" );
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document.write( "Since an = 224, we can say<BR>\n" );
document.write( "an = 7*r^(n-1)<BR>\n" );
document.write( "224 = 7*r^(n-1)<BR>\n" );
document.write( "224/7 = r^(n-1)<BR>\n" );
document.write( "32 = r^(n-1)<BR>\n" );
document.write( "2^5 = r^(n-1)<BR>\n" );
document.write( "2^(6-1) = r^(n-1)\r<BR>\n" );
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document.write( "Matching up terms shows that <BR>\n" );
document.write( "r = 2 and <font color=red size=4>n = 6</font>\r<BR>\n" );
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document.write( "So because n is a whole number, this means we do have a geometric sequence. In this case,\r<BR>\n" );
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document.write( "an = 7*r^(n-1) <BR>\n" );
document.write( "an = 7*(2)^(n-1)<BR>\n" );
document.write( "a1 = 7*2^(1-1) = 7<BR>\n" );
document.write( "a2 = 7*2^(2-1) = 14<BR>\n" );
document.write( "a3 = 7*2^(3-1) = 28<BR>\n" );
document.write( "a4 = 7*2^(4-1) = 56<BR>\n" );
document.write( "a5 = 7*2^(5-1) = 112<BR>\n" );
document.write( "a6 = 7*2^(6-1) = 224\r<BR>\n" );
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document.write( "Adding up the six terms leads to<BR>\n" );
document.write( "a1+a2+a3+a4+a5+a6 = 7+14+28+56+112+224 = 441<BR>\n" );
document.write( "which confirms the answer.\r<BR>\n" );
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document.write( "Or you can use the shortcut formula<BR>\n" );
document.write( "Sn = a1*(1-r^n)/(1-r)<BR>\n" );
document.write( "Sn = 7*(1-2^6)/(1-2)<BR>\n" );
document.write( "Sn = 7*(1-64)/(1-2)<BR>\n" );
document.write( "Sn = 7*(-63)/(-1)<BR>\n" );
document.write( "Sn = 7*63<BR>\n" );
document.write( "Sn = 441<BR>\n" );
document.write( "which is another way to confirm the answer.<BR>\n" );
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