document.write( "Question 1066380: The sum of the fifth and seventh terms of an arithmetic series is 38,while the sum of the first fifteen terms is 375.
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Algebra.Com's Answer #681552 by ikleyn(52795) $\"\"$ $\"About$

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\n" ); document.write( "The sum of the fifth and seventh terms of an arithmetic series is 38,while the sum of the first fifteen terms is 375.
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document.write( "1.   = .   (1)\r\n" );
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document.write( "    Indeed,  = ,\r\n" );
document.write( "             = .\r\n" );
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document.write( "    Adding, you get (1).\r\n" );
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document.write( "    It is a general (and characteristic) property of any arithmetic progression:\r\n" );
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document.write( "            for any three consecutive terms  ,   and    the middle term is equal to the half-sum of its neighbors.\r\n" );
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document.write( "            See the lesson One characteristic property of arithmetic progressions in this site.\r\n" );
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document.write( "2.   = .\r\n" );
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document.write( "    It is easy to prove expressing the terms as neighbors of the central term :\r\n" );
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document.write( "     = ,     = ,\r\n" );
document.write( "     = ,    = ,\r\n" );
document.write( "     = ,    = ,\r\n" );
document.write( "     = ,    = ,\r\n" );
document.write( "     = ,    = ,\r\n" );
document.write( "     = ,    = ,\r\n" );
document.write( "     = ,    = ,\r\n" );
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document.write( "and then adding the terms.\r\n" );
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document.write( "    Hense,  =  = 25.     (2)\r\n" );
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document.write( "3.  Thus from (1) we have  =  = 19.   From (2),   = 25.\r\n" );
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document.write( "    From this, we immediately have 2d =  = 25-19 = 6.\r\n" );
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document.write( "    Hence, the common difference d = 6/2 = 3.\r\n" );
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document.write( "4.  Then  =  = 19 - 5*3 = 19 - 15 = 4.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Answer. $\"a%5B1%5D\"$ = 4; d = 3.\r
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\n" ); document.write( "\n" ); document.write( "The lesson to learn from this solution: Sometimes, it is useful to think,
\n" ); document.write( "                    whether is it possible to build the solution around the properties of the central term of an arithmetic progression.\r
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\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( "    - Mathematical induction and 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
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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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