document.write( "Question 1111116: 2,12,36,80,150,? Series comoletion
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Algebra.Com's Answer #726153 by math_helper(2461)\"\" \"About 
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\n" ); document.write( "n a_n diff diff_2 diff_3
\n" ); document.write( "1 2
\n" ); document.write( "2 12 10
\n" ); document.write( "3 36 24 14
\n" ); document.write( "4 80 44 20 6
\n" ); document.write( "5 150 70 26 6 \r
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\n" ); document.write( "\n" ); document.write( "Because the differences don't reach a constant for 3 levels, the form will be\r
\n" ); document.write( "\n" ); document.write( "\"+a%5Bn%5D+=+k%5B3%5Dn%5E3+%2B+k%5B2%5Dn%5E2+%2B+k%5B1%5Dn+%2B+k%5B0%5D+\"\r
\n" ); document.write( "\n" ); document.write( "Using the first four values, you get these 4 equations:
\n" ); document.write( " \"+k%5B3%5D++%2B+k%5B2%5D+%2B+k%5B1%5D+%2B+k%5B0%5D+=+2+\"
\n" ); document.write( " \"+8k%5B3%5D++%2B+4k%5B2%5D+%2B+2k%5B1%5D+%2B+k%5B0%5D+=+12+\"
\n" ); document.write( " \"+27k%5B3%5D++%2B+9k%5B2%5D+%2B+3k%5B1%5D+%2B+k%5B0%5D+=+36+\"
\n" ); document.write( " \"+64k%5B3%5D++%2B+16k%5B2%5D+%2B+4k%5B1%5D+%2B+k%5B0%5D+=+80+\"
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\n" ); document.write( "Which results in the following matrix equation:\r
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\n" ); document.write( "\n" ); document.write( "Which can be solved using matrix inversion to find:
\n" ); document.write( "\"+k%5B3%5D+=+1+\"
\n" ); document.write( "\"+k%5B2%5D+=+1+\"
\n" ); document.write( "\"+k%5B1%5D+=+0+\"
\n" ); document.write( "\"+k%5B0%5D+=+0+\"\r
\n" ); document.write( "\n" ); document.write( "Therefore \"+highlight%28++a%5Bn%5D+=+n%5E3+%2B+n%5E2++%29\"\r
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\n" ); document.write( " n=1: \"+a%5Bn%5D+=+1%5E3+%2B+1%5E3+=+2+\" (ok)
\n" ); document.write( " n=5: \"+a%5Bn%5D+=+5%5E3+%2B+5%5E2+=+125+%2B+25+=+150+\" (ok)\r
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