document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=87108>Question 87108</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This is a long one I know, can someone please help me with this one?\r<BR>\n" );
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document.write( "Use the arithmetic sequence of numbers 2, 4, 6, 8, 10… to find the following:<BR>\n" );
document.write( "a) 	What is d, the difference between any two consecutive terms? <BR>\n" );
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document.write( "b) 	Using the formula for the nth term of an arithmetic sequence, what is 101st term? Answer: <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? <BR>\n" );
document.write( "Answer:  \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?<BR>\n" );
document.write( "Answer:  <BR>\n" );
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document.write( "e) 	What observation can you make about the successive partial 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.)?<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 #63064 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=63064\">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 2 to 4, 4 to 6, 6 to 8, and 8 to 10 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 10: I'm going to subtract 8 from it (which is the previous term) to get a difference of 2. If I pick 8, and subtract 6, I get a difference of 2. \r<BR>\n" );
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document.write( "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 2 (the 1st term) and I go to 4, this means I increase by 2 each term. If I let n=0 then the term is 2, and if I let n=1 then the term is 4. This basically tells me that the arithmetic sequence is 2n+2. To verify, simply plug in the 1st term (n=0) and you'll get 2. Plug in the 2nd term (n=1) you'll get 4, if I let n=2 I get 6, etc. If I wanted to know the 101st term, let n=100 (remember 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%2B2=202 ALIGN=MIDDLE  ALT=\"2%2Ahighlight%28100%29%2B2=202\"   DISABLED_style= \"max-width:90%; height:auto;\" > So the 101st term is 202\r<BR>\n" );
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document.write( "c)Now lets find the sum of the first 20 terms\r<BR>\n" );
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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\r<BR>\n" );
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document.write( "Since we know the first term is 2, we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D=2 ALIGN=MIDDLE  ALT=\"a%5B1%5D=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "So lets calculate the 20th term. \r<BR>\n" );
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document.write( "Let n=19(remember we started at n=0)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B19%5D=2%2819%29%2B2=38%2B2=40 ALIGN=MIDDLE  ALT=\"a%5B19%5D=2%2819%29%2B2=38%2B2=40\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=40 ALIGN=MIDDLE  ALT=\"a%5Bn%5D=40\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now lets evaluate the sum\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s=%2820%2F2%29%2A%282%2B40%29 ALIGN=MIDDLE  ALT=\"s=%2820%2F2%29%2A%282%2B40%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%2842%29 ALIGN=MIDDLE  ALT=\"s=%2810%29%2A%2842%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=420 ALIGN=MIDDLE  ALT=\"s=420\"   DISABLED_style= \"max-width:90%; height:auto;\" > So the sum of the first 20 terms is 420.\r<BR>\n" );
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document.write( "d)Now lets find the sum of the first 30 terms\r<BR>\n" );
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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 30th term), and n is the number of terms\r<BR>\n" );
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document.write( "Since we know the first term is 2, we know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D=2 ALIGN=MIDDLE  ALT=\"a%5B1%5D=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "So lets calculate the 30th term. \r<BR>\n" );
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document.write( "Let n=29(remember we started at n=0)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B19%5D=2%2829%29%2B2=38%2B2=60 ALIGN=MIDDLE  ALT=\"a%5B19%5D=2%2829%29%2B2=38%2B2=60\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the term <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=60 ALIGN=MIDDLE  ALT=\"a%5Bn%5D=60\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now lets evaluate the sum\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s=%2830%2F2%29%2A%282%2B60%29 ALIGN=MIDDLE  ALT=\"s=%2830%2F2%29%2A%282%2B60%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%2862%29 ALIGN=MIDDLE  ALT=\"s=%2815%29%2A%2862%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=930 ALIGN=MIDDLE  ALT=\"s=930\"   DISABLED_style= \"max-width:90%; height:auto;\" > So the sum of the first 30 terms is 930.\r<BR>\n" );
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document.write( "e)\r<BR>\n" );
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document.write( "Sum of the first 2 terms <BR>\n" );
document.write( "2+4=6 <BR>\n" );
document.write( "Sum of the first 3 terms <BR>\n" );
document.write( "2+4+6=12 <BR>\n" );
document.write( "Sum of the first 4 terms <BR>\n" );
document.write( "2+4+6+8=20 <BR>\n" );
document.write( "Sum of the first 5 terms <BR>\n" );
document.write( "2+4+6+8+10=30 <BR>\n" );
document.write( "Sum of the first 6 terms <BR>\n" );
document.write( "2+4+6+8+10+12=42 <BR>\n" );
document.write( "Sum of the first 7 terms <BR>\n" );
document.write( "2+4+6+8+10+12+14=56 <BR>\n" );
document.write( "Sum of the first 8 terms <BR>\n" );
document.write( "2+4+6+8+10+12+14+16=72 <BR>\n" );
document.write( "Sum of the first 9 terms <BR>\n" );
document.write( "2+4+6+8+10+12+14+16+18=90 <BR>\n" );
document.write( "Sum of the first 10 terms <BR>\n" );
document.write( "2+4+6+8+10+12+14+16+18+20=110\r<BR>\n" );
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document.write( "Notice how the first  sum (which is 6) can be factored like this:<BR>\n" );
document.write( "6=2*3\r<BR>\n" );
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document.write( "Notice how the second  sum (which is 12) can be factored like this:<BR>\n" );
document.write( "12=3*4\r<BR>\n" );
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document.write( "Notice how the third  sum (which is 20) can be factored like this:<BR>\n" );
document.write( "20=4*5\r<BR>\n" );
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document.write( "Notice how the fourth  sum (which is 30) can be factored like this:<BR>\n" );
document.write( "30=5*6\r<BR>\n" );
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document.write( "Notice how the fifth  sum (which is 42) can be factored like this:<BR>\n" );
document.write( "42=6*7\r<BR>\n" );
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document.write( "Notice how the sixth  sum (which is 56) can be factored like this:<BR>\n" );
document.write( "56=7*8\r<BR>\n" );
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document.write( "Notice how the seventh  sum (which is 72) can be factored like this:<BR>\n" );
document.write( "72=8*9\r<BR>\n" );
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document.write( "Notice how the eighth  sum (which is 90) can be factored like this:<BR>\n" );
document.write( "90=9*10\r<BR>\n" );
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document.write( "Notice how the ninth  sum (which is 110) can be factored like this:<BR>\n" );
document.write( "110=10*11\r<BR>\n" );
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document.write( "and this pattern continues...\r<BR>\n" );
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document.write( "Basically, every sum of this sequence can be factored into two numbers in which the second number is one more than the first number\r<BR>\n" );
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document.write( "So each sum follows the sequence <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28n%2B1%29 ALIGN=MIDDLE  ALT=\"n%28n%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >   \r<BR>\n" );
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document.write( "Notice if we let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=2 ALIGN=MIDDLE  ALT=\"n=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%282%2B1%29=2%2A3=6 ALIGN=MIDDLE  ALT=\"2%282%2B1%29=2%2A3=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "which is one of our sums. \r<BR>\n" );
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document.write( "Notice if we let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=3 ALIGN=MIDDLE  ALT=\"n=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > we get\r<BR>\n" );
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document.write( "which is also one of the sums\r<BR>\n" );
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document.write( "So this helps verify our sequence </SPAN>\n" );
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