document.write( "Question 1200515: Find the arithmetic sequence which has the sum of its n terms equal to 2n^2+3n
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Algebra.Com's Answer #834665 by ikleyn(52785) $\"\"$ $\"About$

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document.write( "To solve this problem, you need to know that \r\n" );
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document.write( "    |    the sum of any arithmetic progression is        |\r\n" );
document.write( "    |      a quadratic function of the index \"n\",        | \r\n" );
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document.write( "and vice versa,\r\n" );
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document.write( "    +----------------------------------------------------+\r\n" );
document.write( "    |   any quadratic function of the integer index \"n\"  |\r\n" );
document.write( "    |      creates a unique arithmetic progression.      |\r\n" );
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document.write( "Based on these facts, it is enough to find the first term of the progression\r\n" );
document.write( "and the second term, and then calculate its common difference.\r\n" );
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document.write( "    n= 1:   = 2*1^2+ 3*1 = 5.      = 5.\r\n" );
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document.write( "    n= 2:   = 2*2^2+ 3*2 = 14.     =  -  = 14 - 5 = 9.\r\n" );
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document.write( "This arithmetic progression has the first term   = 5  and the common difference d=  -  = 9 - 5 = 4.\r\n" );
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document.write( "The progression is  5, 9, 13, 17, . . . \r\n" );
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\n" ); document.write( "\n" ); document.write( "On these properties of arithmetic progression, learn from the lessons\r
\n" ); document.write( "\n" ); document.write( "    - Free fall and arithmetic progressions \r
\n" ); document.write( "\n" ); document.write( "    - Uniformly accelerated motions and arithmetic progressions\r
\n" ); document.write( "\n" ); document.write( "    - Increments of a quadratic function form an arithmetic progression\r
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