document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=969019>Question 969019</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sum of tue 2nd and 3rd terme of a gp is12.  The sum of the 3rd and 4th term is<BR>\n" );
document.write( " -36. Find the 1st term of the common ratio\r<BR>\n" );
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document.write( "the geometric sequence of the sum of the 4th and 5th terms is 108 and the sum of the 5th and 6th is 324. Cal.THE common ratio and the value og the 1st term<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #592045 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=592045\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> an = a1 * r^(n-1)\r<BR>\n" );
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document.write( "that's the formula for a geometric series.\r<BR>\n" );
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document.write( "in a geometric series, you get:\r<BR>\n" );
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document.write( "a1 = a<BR>\n" );
document.write( "a2 = a * r^1<BR>\n" );
document.write( "a3 = a * r^2<BR>\n" );
document.write( "a4 = a * r^3<BR>\n" );
document.write( "a5 = a * r^4<BR>\n" );
document.write( "a6 = a * r^5\r<BR>\n" );
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document.write( "in your first problem, you are given that the sum of the second and third terms in the series is equal to 12 and the sum of the third and fourth terms in the series is -36.\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( "(a * r^1) + (a * r^2) = 12<BR>\n" );
document.write( "(a * r^2) + (a * r^3) = -36\r<BR>\n" );
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document.write( "factor out the a in both eqauations to get:\r<BR>\n" );
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document.write( "a * (r^1 + r^2) = 12<BR>\n" );
document.write( "a * (r^2 + r^3) = -36\r<BR>\n" );
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document.write( "solve for a in both equations to get:\r<BR>\n" );
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document.write( "a = 12 / (r^1 + r^2)<BR>\n" );
document.write( "a = -36 * (r^2 + r^3)\r<BR>\n" );
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document.write( "since a = a, this means that 12 / (r^1 + r^2) = -36 / (r^2 + r^3)\r<BR>\n" );
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document.write( "cross multiply in that new equation to get:\r<BR>\n" );
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document.write( "12 * (r^2 + r^3) = -36 * (r^1 + r^2)\r<BR>\n" );
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document.write( "divide both sides of the equation by 12 and divide both sides of the equation by (r^1 + r^2) to get:\r<BR>\n" );
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document.write( "(r^2 + r^3) / (r^1 + r^2) = -36 / 12\r<BR>\n" );
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document.write( "factor out r^2 in the numerator and r in the denominator to get:\r<BR>\n" );
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document.write( "r^2 * (1 + r) / r * (1 + r) = -36 / 12\r<BR>\n" );
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document.write( "simplify to get r = -3.\r<BR>\n" );
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document.write( "you can now use the value of r to find a.\r<BR>\n" );
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document.write( "a2 = a * r^1<BR>\n" );
document.write( "a3 = a * r^2<BR>\n" );
document.write( "a2 + a3 = 12\r<BR>\n" );
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document.write( "since r = -3, you get:\r<BR>\n" );
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document.write( "a2 = a * -3<BR>\n" );
document.write( "a3 = a * (-3)^2 = a * 9\r<BR>\n" );
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document.write( "you are given that a2 + a3 = 12.\r<BR>\n" );
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document.write( "this means that a * -3 + a * 9 = 12.\r<BR>\n" );
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document.write( "combine like terms to get a * 6 = 12\r<BR>\n" );
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document.write( "solve for a to get a = 2.\r<BR>\n" );
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document.write( "that should be your solution.\r<BR>\n" );
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document.write( "let's see if it works.\r<BR>\n" );
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document.write( "a1 = a becomes a1 = 2<BR>\n" );
document.write( "a2 = a * r^1 becomes 2 * -3 = -6<BR>\n" );
document.write( "a3 = a * r^2 becomes 2 * 9 = 18<BR>\n" );
document.write( "a4 = a * r^3 becomes 2 * -27 = -54<BR>\n" );
document.write( "a5 = a * r^4 ....<BR>\n" );
document.write( "a6 = a * r^5 ....\r<BR>\n" );
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document.write( "a2 + a3 = -6 + 18 = 12 ***** this one is ok.<BR>\n" );
document.write( "a3 + a4 = 18 - 54 = -36 ***** this one is ok as well.\r<BR>\n" );
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document.write( "not sure if there's an easier way to solve it, but the way i used worked, so your first question has been answered satisfactorily.\r<BR>\n" );
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document.write( "actually i think i found an easier way.\r<BR>\n" );
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document.write( "i'll use it to solve the second problem.\r<BR>\n" );
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document.write( "the second problem says.\r<BR>\n" );
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document.write( "sum of the 4th and 5th is 108<BR>\n" );
document.write( "sum of the 5th and 6th is 324\r<BR>\n" );
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document.write( "4th term is ar^3<BR>\n" );
document.write( "5th term is ar^4<BR>\n" );
document.write( "6th term is ar^5\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( "ar^3 + ar^4 = 108<BR>\n" );
document.write( "ar^4 + ar^5 = 324\r<BR>\n" );
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document.write( "these are two equations that need to be solved simultaneously.\r<BR>\n" );
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document.write( "multiply the first eqution by r to get:\r<BR>\n" );
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document.write( "ar^4 + ar^5 = 108r<BR>\n" );
document.write( "ar^4 + ar^5 = 324\r<BR>\n" );
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document.write( "subtract the second equation from the first to get:\r<BR>\n" );
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document.write( "0 = 108r - 324\r<BR>\n" );
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document.write( "add 324 to both sides of the equation to get:\r<BR>\n" );
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document.write( "324 = 108r\r<BR>\n" );
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document.write( "divide both sides of the equation by 108 and solve for r to get:\r<BR>\n" );
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document.write( "r = 324 / 108 = 3\r<BR>\n" );
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document.write( "now that you know r, you should be able to solve for a.\r<BR>\n" );
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document.write( "use your first original equation and replace r with 3 to get:\r<BR>\n" );
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document.write( "ar^3 + ar^4 = 108 becomes:\r<BR>\n" );
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document.write( "a*3^3 + a*3^4 = 108 which becomes:\r<BR>\n" );
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document.write( "27a + 81a = 108 which becomes:\r<BR>\n" );
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document.write( "108a = 108\r<BR>\n" );
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document.write( "solve for a to get a = 1.\r<BR>\n" );
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document.write( "a = 1 and r = 3 should be your solution.\r<BR>\n" );
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document.write( "go back to your original equations and replace a with 1 and r with 3 to get:\r<BR>\n" );
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document.write( "ar^3 + ar^4 = 108<BR>\n" );
document.write( "ar^4 + ar^5 = 324\r<BR>\n" );
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document.write( "become:\r<BR>\n" );
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document.write( "1*3^3 + 1*e^4 = 108 which becomes 27 + 81 = 108 which is true.<BR>\n" );
document.write( "1*3^4 + 1*3^5 = 324 which becomes 81 + 243 = 324 which is also true.\r<BR>\n" );
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document.write( "the value of a = 1 and r = 3 are good.\r<BR>\n" );
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document.write( "there you have it.<BR>\n" );
document.write( "two ways to solve the equations.<BR>\n" );
document.write( "take your pick as to which one you prefer.\r<BR>\n" );
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