document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=83751>Question 83751</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the arithmetic sequence of numbers 1, 3, 5, 7, 9,….to find the following:\r<BR>\n" );
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document.write( "a)     What is d, the difference between any two consecutive terms?\r<BR>\n" );
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document.write( "b)     Using the formula for the nth term of an arithmetic sequence, what is 101st  term?\r<BR>\n" );
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document.write( "c)      Using the formula for the sum of an arithmetic sequence, what is the sum of the first 20 terms?\r<BR>\n" );
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document.write( "d)     Using the formula for the sum of an arithmetic sequence, what is the sum of the first 30 terms?\r<BR>\n" );
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document.write( "e)     What observation can you make about these sums of this sequence (HINT:  It would be beneficial to find a few more sums like the sum of the first 2, then the first 3, etc.)?  Express your observations as a general formula in “n.”\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 #60240 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=60240\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a)<BR>\n" );
document.write( "The difference is the factor between each term. So going from 1 to 3, 3 to 5, 5 to 7, you see that its adding 2 each time. To verify, pick one term and subtract the previous term from it. So lets say I choose 7: I'm going to subtract 5 from it to get a difference of 2. If I pick 5, and subtract 3, I get a difference of 2.So the difference is: d=2\r<BR>\n" );
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document.write( "b)<BR>\n" );
document.write( "Using what we found earlier, I know that the sequence counts up by 2 each term. So if I'm at 1 (the 1st term) and I go to 3, this means I increase by 2 each term. If I let n=0 then the term is 1, and if I let n=1 then the term is 3. This basically tells me that the arithmetic sequence is 2n+1. To verify, simply plug in the 1st term (n=0) and you'll get 1. Plug in the 2nd term (n=1) you'll get 3, if I let n=2 I get 5, etc. If I wanted to know the 101st term, let n=100 (zero is the first term) and it comes to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ahighlight%28100%29%2B1=201 ALIGN=MIDDLE  ALT=\"2%2Ahighlight%28100%29%2B1=201\"   DISABLED_style= \"max-width:90%; height:auto;\" > So the 101st term is 201\r<BR>\n" );
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document.write( "c)<BR>\n" );
document.write( "Using the sum of arithmetic series formula:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s=%28n%2F2%29%2A%28a%5B1%5D%2Ba%5Bn%5D%29 ALIGN=MIDDLE  ALT=\"s=%28n%2F2%29%2A%28a%5B1%5D%2Ba%5Bn%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > a[1]=first term, a[n]=nth term (ending term which is the 20th term), and n is the number of terms<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s=%2820%2F2%29%2A%281%2B39%29 ALIGN=MIDDLE  ALT=\"s=%2820%2F2%29%2A%281%2B39%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in values<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s=%2810%29%2A%2840%29 ALIGN=MIDDLE  ALT=\"s=%2810%29%2A%2840%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >Simplify<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s=400 ALIGN=MIDDLE  ALT=\"s=400\"   DISABLED_style= \"max-width:90%; height:auto;\" > So the sum of the first 20 terms is 400.\r<BR>\n" );
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document.write( "d)<BR>\n" );
document.write( "Again using the same formula<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s=%28n%2F2%29%2A%28a%5B1%5D%2Ba%5Bn%5D%29 ALIGN=MIDDLE  ALT=\"s=%28n%2F2%29%2A%28a%5B1%5D%2Ba%5Bn%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  a[1]=first term, a[n]=nth term (ending term which is the 30th term), and n is the number of terms<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s=%2830%2F2%29%2A%281%2B59%29 ALIGN=MIDDLE  ALT=\"s=%2830%2F2%29%2A%281%2B59%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in values<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s=%2815%29%2A%2860%29 ALIGN=MIDDLE  ALT=\"s=%2815%29%2A%2860%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >Simplify<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s=900 ALIGN=MIDDLE  ALT=\"s=900\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "e)<BR>\n" );
document.write( "Sum of the first 2 terms <BR>\n" );
document.write( "1+3=4 <BR>\n" );
document.write( "Sum of the first 3 terms <BR>\n" );
document.write( "1+3+5=9 <BR>\n" );
document.write( "Sum of the first 4 terms <BR>\n" );
document.write( "1+3+5+7=16 <BR>\n" );
document.write( "Sum of the first 5 terms <BR>\n" );
document.write( "1+3+5+7+9=25 <BR>\n" );
document.write( "Sum of the first 6 terms <BR>\n" );
document.write( "1+3+5+7+9+11=36 <BR>\n" );
document.write( "Sum of the first 7 terms <BR>\n" );
document.write( "1+3+5+7+9+11+13=49 <BR>\n" );
document.write( "Sum of the first 8 terms <BR>\n" );
document.write( "1+3+5+7+9+11+13+15=64 <BR>\n" );
document.write( "Sum of the first 9 terms <BR>\n" );
document.write( "1+3+5+7+9+11+13+15+17=81 <BR>\n" );
document.write( "Sum of the first 10 terms <BR>\n" );
document.write( "1+3+5+7+9+11+13+15+17+19=100 \r<BR>\n" );
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document.write( "Notice how the partial sums are all perfect squares. So the sums follow the sequence <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2 ALIGN=MIDDLE  ALT=\"n%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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