document.write( "Question 1121709: The sum of 12 terms of an arithmetic series is 1212. The first term “a” and the common difference “d” are both whole numbers. Write down the first three terms of the series \n" ); document.write( "
Algebra.Com's Answer #737722 by ikleyn(52790)\"\" \"About 
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\n" ); document.write( "Actually, there are  TENS  arithmetic progressions satisfying the given conditions.\r
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\n" ); document.write( "Comment from student : Can you also help me out with this last question?
\n" ); document.write( "Write down the first three terms of a arithmetic series who’s sum is 1212 The series must have a minimum of 13 terms.
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\n" ); document.write( "\n" ); document.write( "My response :\r
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document.write( "The formula for the sum of the first n terms of an arithmetic progression is\r\n" );
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document.write( "    \"S%5Bn%5D\" = \"%28%28a%5B1%5D%2Ba%5Bn%5D%29%2F2%29%2An\".\r\n" );
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document.write( "So, in your case it should be\r\n" );
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document.write( "    \"%28%28a%5B1%5D%2Ba%5Bn%5D%29%2F2%29%2An\" = 1212,   or   \"%28a%5B1%5D%2Ba%5Bn%5D%29%2An\" = 2424.\r\n" );
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document.write( "Notice that 2424 = 24*101, and 101 is a prime number.\r\n" );
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document.write( "It means that \"n\" must divide 101 or must divide some multiple of 101.\r\n" );
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document.write( "Now the first example of such arithmetic progression is THIS:\r\n" );
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document.write( "    101 terms, each equal to 12.\r\n" );
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document.write( "    It is arithmetic progression consisting of 101 equal terms;  each term is equal to 12; the common difference is 0 (zero).\r\n" );
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document.write( "The second example is \r\n" );
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document.write( "    202 terms, each equal to 6.\r\n" );
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document.write( "    It is arithmetic progression consisting of 202 equal terms;  each term is equal to 6; the common difference is 0 (zero).\r\n" );
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document.write( "Having these two examples, you can easily construct other similar examples with the common difference of 0.\r\n" );
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document.write( "I will help you to construct one more example with the non-zero common difference.\r\n" );
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document.write( "    51-th term is 12;\r\n" );
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document.write( "    50-th term is 11;  52-th term is 13;\r\n" );
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document.write( "    49-th term is 10;  53-th term is 14.\r\n" );
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document.write( "    . . .   and  so  on  . . . . . . . . \r\n" );
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document.write( "    1-st term is 12-50 = - 38;  101-th term is 12+50 = 62.\r\n" );
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document.write( "Notice that each pair of the terms in each line has half the sum of 12 (!)\r\n" );
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document.write( "Thus, the progression starts from -38; has the common difference of 1; the last, 101-th term is 62; the sum of the first 101 terms is\r\n" );
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document.write( "    \"S%5B101%5D\" = \"%28%28a%5B1%5D%2Ba%5B101%5D%29%2F2%29%2A101\" = \"%28%28-38%2B62%29%2F2%29%2A101\" = \"%2824%2F2%29%2A101\" = 12*101 = 1212.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Now, if you think carefully on this example, you will be able to construct your own examples with common differences 2, 3, and so on.\r
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\n" ); document.write( "\n" ); document.write( "On arithmetic progressions, there is a bunch of lessons 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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