document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1035275>Question 1035275</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A) The sum to infinity of a geometric series is 3. When the terms of this progression are squared, a new geometric progression is obtained whose sum to infinity is 1.8. Find the first term and the common ratio of each series.<BR>\n" );
document.write( "B) Express the recurring decimal 0.524 in the form p/q, where p and q are integers with no common factor. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #649933 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=649933\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the first term of the infinite geometric series shown below is equal to a.<BR>\n" );
document.write( "the common ratio of the infinite geometric series shown below is equal to r.\r<BR>\n" );
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document.write( "formula for sum of infinite geometric series is equal to a/(1-r)<BR>\n" );
document.write( "if you square r, then the formula becomes a/(1-r^2)\r<BR>\n" );
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document.write( "you are given that a/(1-r) = 3<BR>\n" );
document.write( "you are given that a/(1-r^2) = 1.8\r<BR>\n" );
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document.write( "solve for a in the first equation to get a = 3 * (1-r) = 3 - 3r<BR>\n" );
document.write( "solve for a in the second equation to get a = 1.8 * (1 - r^2) = 1.8  - 1.8r^2.\r<BR>\n" );
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document.write( "since both expressions on the right side of each equation are equal to a, set them equal to each other to get:<BR>\n" );
document.write( "3 - 3r = 1.8 - 1.8r^2\r<BR>\n" );
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document.write( "add 1.8r^2 to both sides of the equation and subtract 1.8 from both sides of the equation to get:<BR>\n" );
document.write( "3 - 3r + 1.8r^2 - 1.8 = 0\r<BR>\n" );
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document.write( "combine like terms and rearrange the terms in descending order of degree to get:<BR>\n" );
document.write( "1.8r^2 - 3r + 1.2 = 0\r<BR>\n" );
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document.write( "factor this quadratic equation to get:<BR>\n" );
document.write( "r = 2/3 or r = 1\r<BR>\n" );
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document.write( "since 0 < r < 1, then r can't be equal to 1, so r has to be equal to 2/3.\r<BR>\n" );
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document.write( "when r = 2/3, the equation of a = 3 - 3r becomes a = 3 - 3*2/3 which becomes a = 3 - 6/3 which becomes a = 3 - 2 which becomes a = 1.\r<BR>\n" );
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document.write( "the first term of the infinite series is 1.<BR>\n" );
document.write( "the common ratio is 2/3.\r<BR>\n" );
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document.write( "-----\r<BR>\n" );
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document.write( "recurring decimal of .524 is equal to 524 / 999.\r<BR>\n" );
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document.write( "you multiply .524..... by 1000 to get 524.525.....\r<BR>\n" );
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document.write( "1 * .524..... is equal to .524.\r<BR>\n" );
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document.write( "subtract 1 from 1000 to get 999.<BR>\n" );
document.write( "subtract .524..... from 524.524..... to get 524.\r<BR>\n" );
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document.write( "your fraction is 524 / 999.\r<BR>\n" );
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document.write( "that can't be simplified any further.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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