document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1064939>Question 1064939</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how many terms should be added to make the sum 16400 of a gp having the sum of first four terms 200 and fifth term 405?(p.s 405 is not sum of fifth term..its just fifth term) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #680044 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><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/cgi-bin/show-face.mpl?user=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=680044\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the sum of n terms of a geometric progression is<BR>\n" );
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document.write( "S(n) = a * ( (1 - r^n) / (1 - r) ), where S(n) is the sum of n terms, a is the first term, r is the common ratio<BR>\n" );
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document.write( "the nth term of a geometric progression is<BR>\n" );
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document.write( "x(n) = a * r^(n-1)<BR>\n" );
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document.write( "we are given<BR>\n" );
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document.write( "1) S(4) = 200 = a * ((1 - r^4) / (1 - r))<BR>\n" );
document.write( "2) x(5) = 405 = a * r^(5-1) = a * r^4<BR>\n" );
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document.write( "we have two equations in two unknowns<BR>\n" );
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document.write( "solve equation 2 for a<BR>\n" );
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document.write( "a = 405 / r^4<BR>\n" );
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document.write( "substitute for a in equation 1<BR>\n" );
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document.write( "200 = (405 / r^4) * ((1 - r^4) / (1 - r)) <BR>\n" );
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document.write( "200 = (405 - 405r^4 / (r^4 - r^5)<BR>\n" );
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document.write( "200r^4 - 200r^5 = 405 - 405r^4<BR>\n" );
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document.write( "3) 200r^5 - 605r^4 +405 = 0<BR>\n" );
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document.write( "a graph of equation 3 shows us that there are three solutions for r (1, 3, and a negative value approximately -0.9)<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+300%2C+200%2C+-2%2C+4%2C+-5000%2C+3500%2C+200x%5E5+-+605x%5E4+%2B405%29+ ALIGN=MIDDLE  ALT=\"+graph%28+300%2C+200%2C+-2%2C+4%2C+-5000%2C+3500%2C+200x%5E5+-+605x%5E4+%2B405%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "we reject r = 1 since that gives us a sum of 0<BR>\n" );
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document.write( "by equation 2, a = 5<BR>\n" );
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document.write( "to recap we have a = 5 and r = 3<BR>\n" );
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document.write( "now we can solve the problem<BR>\n" );
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document.write( "16400 = 5 * (1 - 3^n) / (1 - 3)<BR>\n" );
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document.write( "divide both sides of = by 5<BR>\n" );
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document.write( "3280 = (1 - 3^n) / (-2)<BR>\n" );
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document.write( "1 - 3^n = -6560<BR>\n" );
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document.write( "-3^n = -6561<BR>\n" );
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document.write( "3^n = 6561<BR>\n" );
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document.write( "use definition of logarithm<BR>\n" );
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document.write( "n = log(3) 6561<BR>\n" );
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document.write( "n = 8<BR>\n" );
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document.write( "The sum of the first 8 terms equals 16400<BR>\n" );
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document.write( "to check this we calculate<BR>\n" );
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document.write( "5 * (1 - 3^8) / (1 - 3) =<BR>\n" );
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document.write( "5 * -6560 / -2 = 16400<BR>\n" );
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document.write( " </SPAN>\n" );
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