document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206121>Question 1206121</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An arithmetic progression has a second term of 8 and a fourth term of 18. Find the least number of terms in which the sum of their progression is greater than 1560 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #843360 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=843360\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "d = common difference<BR>\n" );
document.write( "2nd term = 8<BR>\n" );
document.write( "3rd term = (2nd term) + d = 8 + d<BR>\n" );
document.write( "4th term = (3rd term) + d = (8+d) + d = 8 + 2d<BR>\n" );
document.write( "4th term = 18\r<BR>\n" );
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document.write( "8+2d = 18<BR>\n" );
document.write( "2d = 18-8<BR>\n" );
document.write( "2d = 10<BR>\n" );
document.write( "d = 10/2<BR>\n" );
document.write( "d = 5\r<BR>\n" );
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document.write( "2nd term = 8<BR>\n" );
document.write( "3rd term = 8 + d = 8 + 5 = 13<BR>\n" );
document.write( "4th term = 8 + 2d = 8+2*5 = 18\r<BR>\n" );
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document.write( "In short,<BR>\n" );
document.write( "1st term = 3<BR>\n" );
document.write( "2nd term = 8<BR>\n" );
document.write( "3rd term = 13<BR>\n" );
document.write( "4th term = 18<BR>\n" );
document.write( "etc.<BR>\n" );
document.write( "The pattern is that the units digit alternates 3,8,3,8,...\r<BR>\n" );
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document.write( "a = first term = 3<BR>\n" );
document.write( "d = common difference = 5<BR>\n" );
document.write( "n = number of terms<BR>\n" );
document.write( "Sn = sum of the first n terms of an arithmetic sequence<BR>\n" );
document.write( "Sn = (n/2)*(2a + d*(n-1))<BR>\n" );
document.write( "Sn = (n/2)*(2*3 + 5*(n-1))<BR>\n" );
document.write( "Sn = 2.5n^2 + 0.5n\r<BR>\n" );
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document.write( "Sn > 1560<BR>\n" );
document.write( "2.5n^2 + 0.5n > 1560<BR>\n" );
document.write( "2.5n^2 + 0.5n - 1560 > 0\r<BR>\n" );
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document.write( "Let's solve an adjacent or similar equation<BR>\n" );
document.write( "2.5x^2 + 0.5x - 1560 = 0<BR>\n" );
document.write( "Multiply both sides by 2 to end up with<BR>\n" );
document.write( "5x^2 + x - 3120 = 0\r<BR>\n" );
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document.write( "Then let's use the quadratic formula.<BR>\n" );
document.write( "a = 5, b = 1, c = -3120<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1%2B-sqrt%28%281%29%5E2-4%285%29%28-3120%29%29%29%2F%282%285%29%29 ALIGN=MIDDLE  ALT=\"x+=+%28-1%2B-sqrt%28%281%29%5E2-4%285%29%28-3120%29%29%29%2F%282%285%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1%2B-sqrt%2862401%29%29%2F%2810%29 ALIGN=MIDDLE  ALT=\"x+=+%28-1%2B-sqrt%2862401%29%29%2F%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1%2Bsqrt%2862401%29%29%2F%2810%29 ALIGN=MIDDLE  ALT=\"x+=+%28-1%2Bsqrt%2862401%29%29%2F%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1-sqrt%2862401%29%29%2F%2810%29 ALIGN=MIDDLE  ALT=\"x+=+%28-1-sqrt%2862401%29%29%2F%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1%2B249.801922%29%2F%2810%29 ALIGN=MIDDLE  ALT=\"x+=+%28-1%2B249.801922%29%2F%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1-249.801922%29%2F%2810%29 ALIGN=MIDDLE  ALT=\"x+=+%28-1-249.801922%29%2F%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28248.801922%29%2F%2810%29 ALIGN=MIDDLE  ALT=\"x+=+%28248.801922%29%2F%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-250.801922%29%2F%2810%29 ALIGN=MIDDLE  ALT=\"x+=+%28-250.801922%29%2F%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+24.880192 ALIGN=MIDDLE  ALT=\"x+=+24.880192\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-25.080192 ALIGN=MIDDLE  ALT=\"x+=+-25.080192\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The decimal values are approximate.<BR>\n" );
document.write( "Ignore the negative solution since n > 0\r<BR>\n" );
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document.write( "Since x = 24.880192 is one approximate solution to 2.5x^2 + 0.5x - 1560 = 0, let's try n = 24 to see what happens to the arithmetic sum.<BR>\n" );
document.write( "Sn = 2.5n^2 + 0.5n<BR>\n" );
document.write( "S24 = 2.5*24^2 + 0.5*24<BR>\n" );
document.write( "S24 = 1452<BR>\n" );
document.write( "The sum of the first 24 terms is 1452.<BR>\n" );
document.write( "This is too small since we need something larger than 1560.\r<BR>\n" );
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document.write( "Now try n = 25<BR>\n" );
document.write( "Sn = 2.5n^2 + 0.5n<BR>\n" );
document.write( "S25 = 2.5*25^2 + 0.5*25<BR>\n" );
document.write( "S25 = 1575<BR>\n" );
document.write( "The sum of the first 25 terms is 1575.<BR>\n" );
document.write( "We have cleared over the hurdle.\r<BR>\n" );
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document.write( "Therefore, you need at least 25 terms added up to get an arithmetic sum larger than 1560.<BR>\n" );
document.write( "</font> </SPAN>\n" );
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