document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=535095>Question 535095</A>: <I><SPAN STYLE=\"word-break: break-all;\">  The first term of an arithmetic sequence is 1.  The fifth term is 6.  Find the 2009th term. Please show me all your work.Thanks in advance.\r<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 #351789 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=351789\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The following applies for an arithmetic sequence:<BR>\n" );
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document.write( "If the first term of the sequence is A, the next term in the sequence is A + D where D is the common difference between terms. So we can say:<BR>\n" );
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document.write( "Term #1 = A<BR>\n" );
document.write( "Term #2 = A + D<BR>\n" );
document.write( "Term #3 = Term #2 + D = (A + D) + D = A + 2D<BR>\n" );
document.write( "Term #4 = Term #3 + D = (A + 2D) + D = A + 3D<BR>\n" );
document.write( "Term #5 = Term #4 + D = (A + 3D) + D = A + 4D<BR>\n" );
document.write( "Term #6 = Term #5 + D = (A + 4D) + D = A + 5D<BR>\n" );
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document.write( "And so forth. You may be able to see the pattern. The nth term is found by taking the first term and adding to it the quantity n - 1 times D, the difference between terms. In the above example, if we were given A, the first term, we could have found the 6th term by adding to A the product of (6 - 1) times the Difference. In equation form this could be written as the nth term (call it <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5Bn%5D ALIGN=MIDDLE  ALT=\"T%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is given by:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5Bn%5D+=+A+%2B+%28n+-+1%29%2AD ALIGN=MIDDLE  ALT=\"T%5Bn%5D+=+A+%2B+%28n+-+1%29%2AD\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "That's the equation that we can use to solve this problem. <BR>\n" );
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document.write( "From the problem we were told that the first term in the sequence is 1. So we can substitute 1 for A in our equation. We were also told that we are trying to find the nth term when n is 2009. So we can substitute 2009 for n in the equation.<BR>\n" );
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document.write( "All we are missing is D, the difference between terms. We can find D by subtracting the first term (1) from the 5th term (6) and dividing by 4 (the number of terms between the first and 5th terms and it is one less than the number of the last (5th) term. In this instance it is (6 - 1) divided by 4 = 1.25.<BR>\n" );
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document.write( "As a check we can generate the series for the first 5 terms as follows:<BR>\n" );
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document.write( "Term 1 = 1.00<BR>\n" );
document.write( "Term 2 = 2.25<BR>\n" );
document.write( "Term 3 = 3.50<BR>\n" );
document.write( "Term 4 = 4.75<BR>\n" );
document.write( "Term 5 = 6.00 <---  just as the problem says it should be<BR>\n" );
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document.write( "Note that each term after the first was generated by adding 1.25 to the preceding term. So now we know that D is 1.25 and we can write the equation as:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5B2009%5D+=+1+%2B+%282009+-+1%29%2A1.25 ALIGN=MIDDLE  ALT=\"T%5B2009%5D+=+1+%2B+%282009+-+1%29%2A1.25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Combine the numbers in the parentheses to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5B2009%5D+=+1+%2B+%282008%29%2A1.25 ALIGN=MIDDLE  ALT=\"T%5B2009%5D+=+1+%2B+%282008%29%2A1.25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Multiply out the product term on the right side:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5B2009%5D+=+1+%2B+2510 ALIGN=MIDDLE  ALT=\"T%5B2009%5D+=+1+%2B+2510\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now just do the addition on the right side and you have the answer:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%5B2009%5D+=+2511 ALIGN=MIDDLE  ALT=\"T%5B2009%5D+=+2511\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Hope this helps you to understand how arithmetic series are generated and how to develop and use the equation for finding a term in the series.<BR>\n" );
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