document.write( "Question 1107227: In an arithmetic series, t5=16 and S10=175, find S25 \n" ); document.write( "
Algebra.Com's Answer #722253 by ikleyn(52788)\"\" \"About 
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document.write( "For each arithmetic progression,  the sum of terms equally remoted from the first and the last terms is the constant value.\r\n" );
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document.write( "In other words, if  \"a%5B1%5D\", \"a%5B2%5D\", \"a%5B3%5D\", . . . , \"a%5Bn-1%5D\", \"a%5Bn%5D\" is an arifmetic progression of n terms, then\r\n" );
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document.write( "    \"a%5B1%5D\" + \"a%5Bn%5D\" = \"a%5B2%5D\" + \"a%5Bn-1%5D\" = \"a%5B3%5D\" + \"a%5Bn-2%5D\"  and so on.\r\n" );
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document.write( "For the AP of 10 terms\r\n" );
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document.write( "   \"a%5B1%5D\" + \"a%5B10%5D\" = \"a%5B2%5D\" + \"a%5B9%5D\" = \"a%5B3%5D\" + \"a%5B8%5D\" = . . . = \"a%5B5%5D\" + \"a%5B6%5D\".\r\n" );
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document.write( "Hence, the sum  \"a%5B1%5D\" + \"a%5B2%5D\" + . . . + \"a%5B10%5D\"  is 5 times taken the sum \"a%5B5%5D\" + \"a%5B6%5D\":\r\n" );
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document.write( "    \"a%5B1%5D\" + \"a%5B2%5D\" + . . . + \"a%5B10%5D\" = \"5%2A%28a%5B5%5D+%2B+a%5B6%5D%29\".   (1)\r\n" );
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document.write( "        This formula is PARALLEL (or similar) to the general formula for the sum of an AP:    \"S%5Bn%5D\" = \"%28%28a%5B1%5D%2Ba%5Bn%5D%29%2F2%29%2A%28n%2F2%29\".\r\n" );
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document.write( "From the formula (1) we can find the sum  \"a%5B5%5D%2Ba%5B6%5D\".  It is\r\n" );
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document.write( "    \"a%5B5%5D%2Ba%5B6%5D\" = \"S%5B10%5D%2F5\" = \"175%2F5\" = 35.\r\n" );
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document.write( "We also know that \"a%5B5%5D\" = 16.  \r\n" );
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document.write( "It gives  \"a%5B6%5D\" = 35 - 16 = 19.\r\n" );
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document.write( "Thus two neighbor terms of the AP are 16 and 19; they differ by 3 = 19-16, which is the common difference d of the progression.\r\n" );
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document.write( "Having \"a%5B5%5D\" = 16 and d = 3,  we can find  \r\n" );
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document.write( "\"a%5B1%5D\" = \"a%5B5%5D-4%2Ad\" = 16 - 4*3 = 4,\r\n" );
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document.write( "\"a%5B25%5D\" = \"a%5B1%5D%2B24%2Ad\" = 4 + 24*3 = 76.\r\n" );
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document.write( "Now, to find  \"S%5B25%5D\", apply the general formula\r\n" );
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document.write( "\"S%5B25%5D\" = \"%28%28a%5B1%5D%2Ba%5B25%5D%29%2F2%29%2A25\" = \"%28%284%2B76%29%2F2%29%2A25\" = 1000.\r\n" );
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document.write( "Answer.  \"S%5B25%5D\" = 1000.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\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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\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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