document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192881>Question 1192881</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the 20th term of the sequence 0, 3, 8, 15, 24, 35, 48, . .<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 #824806 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=824806\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I'm assuming that this is a quadratic sequence because the gap between terms 3, 5, 7, 9, 11, 13, ... are of an arithmetic sequence (common difference d = 2). <BR>\n" );
document.write( "If this assumption is not the case, then please let me know and I'll update the solution.\r<BR>\n" );
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document.write( "If it is quadratic, then it means you'll have something of the form<BR>\n" );
document.write( "y = ax^2+bx+c<BR>\n" );
document.write( "where x is the term number and y is the term itself.\r<BR>\n" );
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document.write( "Examples: <BR>\n" );
document.write( "(x,y) = (1,0) means the first term is 0.<BR>\n" );
document.write( "(x,y) = (2,3) means the second term is 3.<BR>\n" );
document.write( "and so on.\r<BR>\n" );
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document.write( "Plug in (x,y) = (1,0) to find that<BR>\n" );
document.write( "y = ax^2+bx+c<BR>\n" );
document.write( "0 = a(1)^2+b(1)+c<BR>\n" );
document.write( "0 = a+b+c<BR>\n" );
document.write( "a+b+c = 0\r<BR>\n" );
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document.write( "Then plug in x = 2 and y = 3 to get<BR>\n" );
document.write( "4a+2b+c = 3\r<BR>\n" );
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document.write( "And plug in x = 3 and y = 8 to get<BR>\n" );
document.write( "9a+3b+c = 8\r<BR>\n" );
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document.write( "The task is to solve this system of equations<BR>\n" );
document.write( "a+b+c = 0<BR>\n" );
document.write( "4a+2b+c = 3<BR>\n" );
document.write( "9a+3b+c = 8<BR>\n" );
document.write( "I'll skip the steps because it's a very long process, but the punchline is that<BR>\n" );
document.write( "a = 1<BR>\n" );
document.write( "b = 0<BR>\n" );
document.write( "c = -1\r<BR>\n" );
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document.write( "This means<BR>\n" );
document.write( "y = ax^2+bx+c<BR>\n" );
document.write( "y = 1x^2+0x+(-1)<BR>\n" );
document.write( "y = x^2-1<BR>\n" );
document.write( "represents a nice simple formula to generate any term (y) based on the term number (x).\r<BR>\n" );
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document.write( "Checking the formula:\r<BR>\n" );
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document.write( "Plug in x = 1 to represent the first term<BR>\n" );
document.write( "y = x^2 - 1<BR>\n" );
document.write( "y = 1^2 - 1<BR>\n" );
document.write( "y = 0<BR>\n" );
document.write( "That works out because 0 is indeed the first term\r<BR>\n" );
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document.write( "Repeat for x = 2<BR>\n" );
document.write( "y = x^2 - 1<BR>\n" );
document.write( "y = 2^2 - 1<BR>\n" );
document.write( "y = 3<BR>\n" );
document.write( "That works out as well\r<BR>\n" );
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document.write( "I'll let you check the other terms.\r<BR>\n" );
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document.write( "Plugging in x = 20 will then lead you to y = 399 which is the 20th term.\r<BR>\n" );
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document.write( "Once again, this all depends on if the sequence given is quadratic in nature.<BR>\n" );
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