document.write( "Question 146054: what is the geometric series \"-9%2F2%2B3%2F2-1%2F2%2B1%2F6\"-...+\"1%2F39366\" in summation notation? \n" ); document.write( "
Algebra.Com's Answer #106617 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
what is the geometric series \"-9%2F2\"+\"3%2F2\"+\"-1%2F2\"+\"1%2F6\"-...+\"1%2F39366\" in summation notation?
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document.write( "\"sum%28a%5B1%5Dr%5E%28k-1%29%2C+k=1%2C+N+%29\"\r\n" );
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document.write( "We divide the 2nd term by the 1st term to find the common ratio \"r\"\r\n" );
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document.write( "To check we divide the 3rd term by the 2nd term to see if we get the \r\n" );
document.write( "same common ratio \"-1%2F3\":\r\n" );
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document.write( "To double check we divide the 4th term by the 3rd term to see if we get the \r\n" );
document.write( "same common ratio \"-1%2F3\":\r\n" );
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document.write( "\"%281%2F6%29%2F%28-1%2F2%29=%281%2F6%29%2A%282%2F%28-1%29%29=2%2F%28-6%29=-1%2F3\"\r\n" );
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document.write( "Now that we are triple-sure that the common ratio \"r=-1%2F3\",\r\n" );
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document.write( "we will use the formula for the nth term:\r\n" );
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document.write( "\"a%5Bn%5D+=+a%5B1%5Dr%5E%28n-1%29\"\r\n" );
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document.write( "to find out how many terms it has:\r\n" );
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document.write( "\"a%5Bn%5D+=+1%2F39366\" is the last, or nth, term:\r\n" );
document.write( "\"a%5B1%5D+=+-9%2F2\" is the first term\r\n" );
document.write( "\"r+=+-1%2F3\" is the common ratio.\r\n" );
document.write( "Substituting:\r\n" );
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document.write( "\"1%2F39366=%28-9%2F2%29%28-1%2F3%29%5E%28n-1%29\"\r\n" );
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document.write( "Multiply both sides by \"%2839366%2F1%29\"\r\n" );
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document.write( "\"%2839366%2F1%29%281%2F39366%29=%2839366%2F1%29%28-9%2F2%29%28-1%2F3%29%5E%28n-1%29\"\r\n" );
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document.write( "\"1=-177147%28-1%2F3%29%5E%28n-1%29\"\r\n" );
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document.write( "Write \"%28-1%2F3%29%5E%28n-1%29\" as \"%28%28-1%29%5E%28n-1%29%2F3%5E%28n-1%29%29\"\r\n" );
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document.write( "\"1=-177147%28%28-1%29%5E%28n-1%29%2F3%5E%28n-1%29%29\"\r\n" );
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document.write( "Multiply both sides by \"3%5E%28n-1%29\"\r\n" );
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document.write( "Observe that \"177147=3%5E11\", so substitute that:\r\n" );
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document.write( "\"3%5E%28n-1%29=%283%5Ell%29%28-1%29%5E%28n-1%29\"\r\n" );
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document.write( "Divide both sides by \"3%5E11\"\r\n" );
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document.write( "\"3%5E%28n-1%29%2F3%5E11=%28-1%29%5E%28n-1%29\"\r\n" );
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document.write( "Subtract exponents on the left:\r\n" );
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document.write( "\"3%5E%28n-1-11%29=%28-1%29%5E%28n-1%29\"\r\n" );
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document.write( "\"3%5E%28n-12%29=%28-1%29%5E%28n-1%29\"\r\n" );
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document.write( "Since the right side is a power of\r\n" );
document.write( "\"-1\", it is either 1 or -1\r\n" );
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document.write( "But no power of 3 can be negative, so\r\n" );
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document.write( "\"3%5E%28n-12%29=1\"\r\n" );
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document.write( "And since the only power of 3 that gives 1\r\n" );
document.write( "is the 0 power, i.e., 30=1, then\r\n" );
document.write( "the exponent n-12 must equal 0.\r\n" );
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document.write( "n-12=0\r\n" );
document.write( "   n=12.\r\n" );
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document.write( "So there are 12 terms.  So \r\n" );
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document.write( "\"sum%28a%5B1%5Dr%5E%28k-1%29%2C+k=1%2C+N+%29\"\r\n" );
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document.write( "becomes:\r\n" );
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document.write( "\"sum%28%28-9%2F2%29%28-1%2F3%29%5E%28k-1%29%2C+k=1%2C+12+%29\"\r\n" );
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document.write( "write \"-9%2F2\" as \"%28%28-1%29%2F1%29%289%2F2%29\"\r\n" );
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document.write( "\"sum%28%28%28-1%29%2F1%29%289%2F2%29%28-1%2F3%29%5E%28k-1%29%2C+k=1%2C+12+%29\"\r\n" );
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document.write( "write \"%28-1%2F3%29%5E%28k-1%29\" as \"%28-1%29%5E%28k-1%29%2F3%5E%28k-1%29\"\r\n" );
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document.write( "Write \"9\" as \"3%5E2\":\r\n" );
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document.write( "\"sum%28%28%28-1%293%5E2%28-1%29%5E%28k-1%29%29%2F%282%283%5E%28k-1%29%29%29%2C+k=1%2C+12+%29\"\r\n" );
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document.write( "Add exponents of \"-1\" on top:\r\n" );
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document.write( "\"sum%28%283%5E2%28-1%29%5E%28k-1%2B1%29%29%2F%282%283%5E%28k-1%29%29%29%2C+k=1%2C+12+%29\"\r\n" );
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document.write( "\"sum%28%283%5E2%28-1%29%5Ek%29%2F%282%283%5E%28k-1%29%29%29%2C+k=1%2C+12+%29\"\r\n" );
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document.write( "Subtract exponents of 3:\r\n" );
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document.write( "\"sum%28%28%28-1%29%5Ek%29%2F%282%283%5E%28k-1-2%29%29%29%2C+k=1%2C+12+%29\"\r\n" );
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document.write( "\"sum%28%28%28-1%29%5Ek%29%2F%282%2A3%5E%28k-3%29%29%2C+k=1%2C+12+%29\"\r\n" );
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document.write( "Edwin
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