document.write( "Question 1134407: Find four numbers in A.P whose sum is 20 and the sum of whose squares is 120. \n" ); document.write( "

Algebra.Com's Answer #751788 by ikleyn(52781) $\"\"$ $\"About$

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document.write( "The four terms of the given AP are nicely (symmetrically) located around their central point x= 5, which is their arithmetic mean  .\r\n" );
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document.write( "I will use this fact in order for to simplify the solution. I also introduce  2d as the common difference of the AP \r\n" );
document.write( "(instead of using the tradition designation d for the common difference).\r\n" );
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document.write( "Then obviously\r\n" );
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document.write( "     = 5 - 3d,\r\n" );
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document.write( "     = 5 - d,\r\n" );
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document.write( "     = 5 + d,\r\n" );
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document.write( "     = 5 + 3d.\r\n" );
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document.write( "The sum of squares of these four terms is\r\n" );
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document.write( "     =  +  +  +  = .\r\n" );
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document.write( "So, for \"d\" you have this equation\r\n" );
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document.write( "    100 + 20d^2 = 120,\r\n" );
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document.write( "from which you get\r\n" );
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document.write( "    d^2 =  = 1;   hence,  d = +/- 1.\r\n" );
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document.write( "Thus the four terms of the AP are  5-3 = 2;  5-1 = 4;  5+1 = 6  and  5+3 = 8.\r\n" );
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document.write( "ANSWER.   The four terms of the AP are  2, 4, 6, 8.\r\n" );
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document.write( "          The reversed sequence  8, 6, 4, 2  is the solution, also.  It corresponds to the value  d= -1.\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( "    - Math Olimpiad level problem on arithmetic progression \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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\n" ); document.write( "\n" ); document.write( "I forced to make this reminder because MANY (if not all) visitors IGNORE THIS RULE. Unfortunately.\r
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\n" ); document.write( "\n" ); document.write( "Which I consider as an extreme degree of discourtesy.\r
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