document.write( "Question 1195430: The first term of an AP is 3. Given that the sum of the first 6 terms is 48 and that the sum of all the terms is 168 .calculate the common difference,the number of terms in the AP and the last term \n" ); document.write( "
Algebra.Com's Answer #827908 by ikleyn(52790)\"\" \"About 
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\n" ); document.write( "The first term of an AP is 3. Given that the sum of the first 6 terms is 48
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document.write( "The sum of the first 6 terms is 48.\r\n" );
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document.write( "This sum is 6 times the average of the first 6 terms\r\n" );
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document.write( "and is 6 times the average of the 1st and 6th terms of this AP\r\n" );
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document.write( "    \"6%2A%28%28a%5B1%5D%2Ba%5B6%5D%29%2F2%29\" = 48,\r\n" );
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document.write( "or\r\n" );
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document.write( "    \"6%2A%283+%2B+a%5B6%5D%29\" = 48*2\r\n" );
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document.write( "    \"18+%2B+6%2Aa%5B6%5D\" = 96\r\n" );
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document.write( "    \"6%2Aa%5B6%5D\" = 96 - 18 = 78\r\n" );
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document.write( "    \"a%5B6%5D\" = 78/6 = 13.\r\n" );
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document.write( "Thus  the 6th term of the AP is 13:  \r\n" );
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document.write( "    \"a%5B6%5D\" = \"a%5B1%5D%2B5%2Ad\" = 13\r\n" );
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document.write( "    3 + 5d = 13\r\n" );
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document.write( "        5d = 13-3 = 10\r\n" );
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document.write( "         d = 10/5 = 2.\r\n" );
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document.write( "Thus the common difference of the AP is  d= 2.\r\n" );
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document.write( "    \r\n" );
document.write( "Next, the sum of all the terms of the AP is 168\r\n" );
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document.write( "    \"S%5Bn%5D\" = \"%282a%5B1%5D+%2B+%28n-1%29d%29%2A%28n%2F2%29\" = 168\r\n" );
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document.write( "or\r\n" );
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document.write( "    (2*3+(n-1)*2)*n = 168*2\r\n" );
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document.write( "    6n + 2*(n-1)*n = 336\r\n" );
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document.write( "    2n^2 + 4n - 336 = 0\r\n" );
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document.write( "     n^2 + 2n - 168 = 0\r\n" );
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document.write( "     (n-12)*(n+14) = 0\r\n" );
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document.write( "The roots of this quadratic equation are 12 and -14; since we are looking for the number of terms n,\r\n" );
document.write( "we see that the solution is the positive root n= 12.\r\n" );
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document.write( "ANSWER.  The common difference of the AP is 2;\r\n" );
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document.write( "         the number of terms is 12, and the last term is  \"a%5B12%5D\" = 3 + 2*(12-1) = 25.\r\n" );
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\n" ); document.write( "\n" ); document.write( "All questions are answered.\r
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\n" ); document.write( "\n" ); document.write( "For introductory lessons on arithmetic progressions see \r
\n" ); document.write( "\n" ); document.write( "    - Arithmetic progressions\r
\n" ); document.write( "\n" ); document.write( "    - The proofs of the formulas for arithmetic progressions \r
\n" ); document.write( "\n" ); document.write( "    - Problems on arithmetic progressions \r
\n" ); document.write( "\n" ); document.write( "    - Word problems on arithmetic progressions\r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic \"Arithmetic progressions\".\r
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\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
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