document.write( "Question 1150799: The interior angles of a convex nonagon have degree measures that are integers in arithmetic sequence. What is the smallest angle possible in this nonagon? \n" ); document.write( "

Algebra.Com's Answer #772256 by ikleyn(52781) $\"\"$ $\"About$

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document.write( "From one side, the sum of interior angles of any 9-gon is  (9-2)*180 = 7*180 degrees.\r\n" );
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document.write( "From the other side, the sum of the first n terms of any arithmetic progression is\r\n" );
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document.write( "     = .\r\n" );
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document.write( "In our case,   = 7*180,  so\r\n" );
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document.write( "     = 140  degrees.\r\n" );
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document.write( "For arithmetic progression, the average of any two terms, equally remoted from the central term, is the same.\r\n" );
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document.write( "In particular, the central term   is 140 degrees.\r\n" );
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document.write( "The terms of the AP are\r\n" );
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document.write( "     = 140 -  d,    = 140 +  d,\r\n" );
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document.write( "     = 140 - 2d,    = 140 + 2d,\r\n" );
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document.write( "     = 140 - 3d,    = 140 + 3d,\r\n" );
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document.write( "     = 140 - 4d,    = 140 + 4d.\r\n" );
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document.write( "To make  as small as possible, we should take the common difference as large as possible.\r\n" );
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document.write( "We have two constraints:   must be positive,   > 0,                                      (1)\r\n" );
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document.write( "and\r\n" );
document.write( "                           must be less than 180 degrees;  so    must be 176 degrees.    (2)\r\n" );
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document.write( "Of these two constraints, the constraint (2) is more cumbersome, and it gives\r\n" );
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document.write( "    d = 36/4 = 9 degrees.\r\n" );
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document.write( "Then both constraints (1) and (2) are satisfied.\r\n" );
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document.write( "Thus the minimum angle is   = 140 - 4*9 = 140 - 36 = 104 degrees.    ANSWER\r\n" );
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