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\n" ); document.write( "Since the sum of the four numbers and the sum of their squares are both whole numbers, the four numbers are probably whole numbers. So solve the problem using logical trial and error.
\n" ); document.write( "The sum of the four numbers is 20, so their average is 5. The easiest way to get and AP of four whole numbers with an average of 5 is 2, 4, 6, 8. (The AP 5, 5, 5, 5 is easier, but not very interesting; and it doesn't satisfy the other condition of the problem.)
\n" ); document.write( "And the AP 2, 4, 6, 8 does satisfy the other condition: 2^2+4^2+6^2+8^2 = 4+16+36+64 = 120.
\n" ); document.write( "So the AP is 2, 4, 6, 8.
\n" ); document.write( "If you need an algebraic solution, here is one that makes the algebra relatively easy.
\n" ); document.write( "Since the average of the four numbers is 5, and since we are guessing that the numbers are whole numbers, let the middle two numbers be 5-a and 5+a. That makes the common difference 2a, so the other two numbers are 5-3a and 5+3a.
\n" ); document.write( "The sum of the squares of those four numbers is
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\n" ); document.write( "Since the sum of the squares is 120,
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\n" ); document.write( "And so the four numbers are
\n" ); document.write( "5-3a = 2
\n" ); document.write( "5-a = 4
\n" ); document.write( "5+a = 6
\n" ); document.write( "5+3a = 8
\n" ); document.write( "Note when we got to the point where we had a^2=1, we could have chosen a = -1; we would have ended up with the same four numbers:
\n" ); document.write( "5-3a = 8
\n" ); document.write( "5-a = 6
\n" ); document.write( "5+a = 4
\n" ); document.write( "5+3a = 2
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