document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1167088>Question 1167088</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The differences between the terms of the geometric sequence are known: '[a2-a1=-6,a3-a2=12' Find the sum of the first ten terms of that sequence. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #791663 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=791663\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> it looks like the differences are doubling every time and the signs are reversing every time.\r<BR>\n" );
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document.write( "A2 minus A1 = -6<BR>\n" );
document.write( "A3 minus A2 = 12\r<BR>\n" );
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document.write( "since this is a geometric series, the formula is An = A1 * r^(n-1), where r is the common ratio.\r<BR>\n" );
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document.write( "this means that:\r<BR>\n" );
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document.write( "A2 = A1 * r<BR>\n" );
document.write( "A3 = A1 * r^2\r<BR>\n" );
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document.write( "we also know that:\r<BR>\n" );
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document.write( "A2 - A1 = -6<BR>\n" );
document.write( "A3 - A2 = 12\r<BR>\n" );
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document.write( "since A2 = A1 * r, then A2 - A1 becomes A1 * r - A1 = -6\r<BR>\n" );
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document.write( "since A3 = A1 * r^2, then A3 - A2 = 12 becomes A1 * r^2 - A1 * r = 12\r<BR>\n" );
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document.write( "factor out the A1 in both equations to get:\r<BR>\n" );
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document.write( "A1 * (r - 1) = -6<BR>\n" );
document.write( "A1 * (r^2 - r) = 12\r<BR>\n" );
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document.write( "solve for A1 in both equations to get:\r<BR>\n" );
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document.write( "A1 = -6 / (r - 1)<BR>\n" );
document.write( "A1 = 12 / (r^2 - r)\r<BR>\n" );
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document.write( "since they are both equal to A1, we get:\r<BR>\n" );
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document.write( "-6 / (r - 1) = 12 / (r^2 - r)\r<BR>\n" );
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document.write( "cross multiply to get:\r<BR>\n" );
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document.write( "-6 * (r^2 - r) = 12 * (r - 1)\r<BR>\n" );
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document.write( "divide both sides of this equation by (r - 1) to get:\r<BR>\n" );
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document.write( "-6 * (r^2 - r) / (r - 1) = 12\r<BR>\n" );
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document.write( "divide both sides of this equation by -6 to get:\r<BR>\n" );
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document.write( "(r^2 - r) / (r - 1) = 12 / -6 = -2\r<BR>\n" );
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document.write( "factor the numerator on the left side of the equation to get:\r<BR>\n" );
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document.write( "r * (r - 1) / (r - 1) = -2\r<BR>\n" );
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document.write( "simplify the equation to get:\r<BR>\n" );
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document.write( "r = -2\r<BR>\n" );
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document.write( "you now know that the common ratio = -2.\r<BR>\n" );
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document.write( "knowing that, we get:\r<BR>\n" );
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document.write( "A2 - A1 = -6 becomes A1 * r - A1 = -6 which becomes A1 * -2 - A1 = -6  which becomes:\r<BR>\n" );
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document.write( "-2 * A1 - A1 = -6\r<BR>\n" );
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document.write( "combine like terms to get -3 * A1 = -6\r<BR>\n" );
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document.write( "solve for A1 to get A1 = -6 / -3 = 2\r<BR>\n" );
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document.write( "since we now know the value of A1, we get:\r<BR>\n" );
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document.write( "A1 = 2<BR>\n" );
document.write( "A2 = 2 * -2 = -4<BR>\n" );
document.write( "A3 = 2 * (-2)^2 = 2 * 4 = 8\r<BR>\n" );
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document.write( "we have:\r<BR>\n" );
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document.write( "A1 = 2<BR>\n" );
document.write( "A2 = -4<BR>\n" );
document.write( "A3 = 8\r<BR>\n" );
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document.write( "A2 - A1 = -4 - 2 = -6<BR>\n" );
document.write( "A3 - A2 = 8 - (-4) = 12\r<BR>\n" );
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document.write( "it looks like we have A1 and the common ratio of -2 nailed down pretty good and we can proceed with finding the first ten terms of our sequence.\r<BR>\n" );
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document.write( "we get:\r<BR>\n" );
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document.write( "A1 = 2<BR>\n" );
document.write( "A2 = 2 * -2 = -4<BR>\n" );
document.write( "A3 = -4 * -2 = 8<BR>\n" );
document.write( "A4 = 8 * -2 = -16<BR>\n" );
document.write( "A5 = -16 * -2 = 32<BR>\n" );
document.write( "A6 = 32 * -2 = -64<BR>\n" );
document.write( "A7 = -64 * -2 = 128<BR>\n" );
document.write( "A8 = 128 * -2 = -256<BR>\n" );
document.write( "A9 = -256 * -2 = 512<BR>\n" );
document.write( "A10 = 512 * -2 = -1024\r<BR>\n" );
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document.write( "we add all ten terms to get:\r<BR>\n" );
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document.write( "the sum of the first 10 terms of the sequence = -684 !!!!!\r<BR>\n" );
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document.write( "we can also use the sum of the terms of a geometric sequence to see if we get the same value.\r<BR>\n" );
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document.write( "that formula is:\r<BR>\n" );
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document.write( "Sn = A1 * (r - r^n) / (1 - r)\r<BR>\n" );
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document.write( "when n = 10 and A1 = 2, this formula becomes:\r<BR>\n" );
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document.write( "S10 = 2 * (-2 - (-2)^10) / (1 - (-2))\r<BR>\n" );
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document.write( "simplify this formula to get:\r<BR>\n" );
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document.write( "S10 = 2 * (-2 - 1024) / (-3)\r<BR>\n" );
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document.write( "simplify further to get:\r<BR>\n" );
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document.write( "S10 = -2 * -1026 / -3\r<BR>\n" );
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document.write( "solve for S10 to get:\r<BR>\n" );
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document.write( "S10 = -684 !!!!!\r<BR>\n" );
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document.write( "the formula gives the same answer as manually summing up the terms.\r<BR>\n" );
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document.write( "this confirms that the sum formula works and that the sum of the first ten terms of the sequence is -684.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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