document.write( "Question 1140687: In an AP consisting of 15 terms, the sum of the last five terms is 305. If the sixth term is 26, find the sum of this AP \n" ); document.write( "
Algebra.Com's Answer #761219 by ikleyn(52790)\"\" \"About 
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document.write( "The last five terms of this progression are  \"a%5B11%5D\",  \"a%5B12%5D\",  \"a%5B13%5D\",   \"a%5B14%5D\"  and  \"a%5B15%5D\".\r\n" );
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document.write( "The middle term of these 5 terms is  \"a%5B13%5D\", and it is equal to   \"1%2F5\"  of the sum of that five terms.\r\n" );
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document.write( "     It is the key point in the solution, and you should clearly understand it:\r\n" );
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document.write( "          the average of any odd number of sequential terms of an AP is  the middle term of this partial sequence.\r\n" );
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document.write( "Thus  \"a%5B13%5D\" = \"305%2F5\" = 61.\r\n" );
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document.write( "Now we know  \"a%5B13%5D\" = 61  and  \"a%5B6%5D\" = 26.\r\n" );
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document.write( "Between  \"a%5B13%5D\" and  \"a%5B6%5D\",  there are 13-6 = 7 gaps, each of which is equal to the common difference of the AP.\r\n" );
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document.write( "Hence, the common difference  d = \"%2861-26%29%2F7\" = \"35%2F7\" = 5.\r\n" );
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document.write( "Then the first term is  \"a%5B1%5D\" = \"a%5B6%5D\" - 5*5 = 26 - 25 = 1.\r\n" );
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document.write( "     the 15-th term is  \"a%5B15%5D\" = \"a%5B1%5D\" + (15-1)*d = 1 + 14*5 = 71.\r\n" );
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document.write( "Now the sum of the 15 terms of the AP is  \"%28%28a%5B1%5D%2Ba%5B15%5D%29%2F2%29%2A15\" = \"%28%281%2B71%29%2F2%29%2A15\" = 540.\r\n" );
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\n" ); document.write( "\n" ); document.write( "There is a bunch of lessons on arithmetic progressions in this site:\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( "    - One characteristic property of arithmetic progressions\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on arithmetic progressions \r
\n" ); document.write( "\n" ); document.write( "    - Calculating partial sums of arithmetic progressions \r
\n" ); document.write( "\n" ); document.write( "    - Mathematical induction and arithmetic progressions\r
\n" ); document.write( "\n" ); document.write( "    - Mathematical induction for sequences other than arithmetic or geometric\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( "Save the link to this textbook together with its description\r
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\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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