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4,19,58,133,256
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\n" ); document.write( "we solve this by taking successive levels of differences between succeeding terms - this is called \"The method of common differences\"
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\n" ); document.write( "1) 15, 39, 75, 123
\n" ); document.write( "2) 24, 36, 48
\n" ); document.write( "3) 12, 12
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\n" ); document.write( "we see that level 3 has the same difference between the 3 terms, so we have a n^3 term
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\n" ); document.write( "the general form for a cubic is
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\n" ); document.write( "an^3 + bn^2 + cn + d, where a,b,c,d are numbers
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\n" ); document.write( "we know that
\n" ); document.write( "a(1^3) + b(1^2) + c(1) + d = 4
\n" ); document.write( "4) a + b + c + d = 4
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\n" ); document.write( "a(2^3) + b(2^2) + c(2) + d = 19
\n" ); document.write( "5) 8a + 4b +2c + d = 19
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\n" ); document.write( "a(3^3) + b(3^2) + c(3) + d = 58
\n" ); document.write( "6) 27a + 9b +3c +d = 58
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\n" ); document.write( "a(4^3) + b(4^2) + c(4) + d = 133
\n" ); document.write( "7) 64a +16b +4c + d = 133
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\n" ); document.write( "Equations 4 through 7 gives us 4 equations in 4 unknowns which can be solved using any n by n solver for linear systems
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\n" ); document.write( "a=2, b=0, c=1, d=1
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\n" ); document.write( "a(n) = 2n^3 + n + 1
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\n" ); document.write( "a(7) = 2(7^3) + 7 + 1 = 694
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