document.write( "Question 874586: what is the pattern in this sequence: 1, 10, 25, 55, 109, 196, and 325 \n" ); document.write( "
Algebra.Com's Answer #527625 by KMST(5328)\"\" \"About 
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The differences between consecutive terms are
\n" ); document.write( "9, 15, 30, 54, 87, 129.
\n" ); document.write( "If they were the same, it would be an arithmetic sequence,
\n" ); document.write( "and calling the initial term \"x%5B0%5D\" , and the common difference \"d\" ,
\n" ); document.write( "term number \"n\" would be
\n" ); document.write( "\"x%5Bn%5D=x%5B0%5D%2Bn%2Ad\" .
\n" ); document.write( "In other words, \"x%5Bn%5D\" would be a linear function of \"n\" .
\n" ); document.write( "
\n" ); document.write( "The differences of the differences are
\n" ); document.write( "\"15-9=6\"
\n" ); document.write( "\"30-15=15\"
\n" ); document.write( "\"54-30=24\"
\n" ); document.write( "\"87-54=33\"
\n" ); document.write( "\"129-87=42\"
\n" ); document.write( "If those differences were the same, \"x%5Bn%5D\" would be a quadratic function of \"n\" .
\n" ); document.write( "
\n" ); document.write( "The differences of the differences of the differences are all \"9\" :
\n" ); document.write( "\"15-6=9\"
\n" ); document.write( "\"24-15=9\"
\n" ); document.write( "\"33-24=9\"
\n" ); document.write( "\"42-33=9\"
\n" ); document.write( "\"x%5Bn%5D\" must be a cubic function of \"n\" .
\n" ); document.write( "
\n" ); document.write( "From the differences, we can make a table and figure out that the next terms are
\n" ); document.write( "505, 745, 1054.
\n" ); document.write( "The table would be
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "Finding a formula is a little harder.
\n" ); document.write( "The fact that the third differences are constant means that \"x%5Bn%5D\" is a cubic function of \"n\"
\n" ); document.write( "\"x%5Bn%5D=a%2An%5E3%2Bb%2An%5E2%2Bc%2An%2Bd\"
\n" ); document.write( "\"x%5B0%5D=d=1\"
\n" ); document.write( "\"x%5B1%5D=a%2Bb%2Bc%2B1=10\"-->\"a%2Bb%2Bc=9\"
\n" ); document.write( "\"x%5B2%5D=a%2A2%5E3%2Bb%2A2%5E2%2Bc%2A2%2B1=25\"-->\"8a%2B4b%2B2c%2B1=25\"-->\"8a%2B4b%2B2c=24\"-->\"4a%2B2b%2Bc=12\"
\n" ); document.write( "\"x%5B3%5D=a%2A3%5E3%2Bb%2A3%5E2%2Bc%2A3%2B1=55\"-->\"27a%2B9b%2B3c%2B1=55\"-->\"27a%2B9b%2B3c=54\"-->\"9a%2B3b%2Bc=18\"
\n" ); document.write( "Solving \"system%28a%2Bb%2Bc=9%2C4a%2B2b%2Bc=12%2C9a%2B3b%2Bc=18%29\" we find can find the formula easily enough:
\n" ); document.write( "From \"system%28a%2Bb%2Bc=9%2C4a%2B2b%2Bc=12%2C9a%2B3b%2Bc=18%29\" ,
\n" ); document.write( "subtracting the first equation from the other two we get
\n" ); document.write( "\"system%28a%2Bb%2Bc=9%2C3a%2Bb=3%2C8a%2B2b=9%29\"
\n" ); document.write( "Subtracting 2 times \"3a%2Bb=3\" , or \"6a%2B2b=6\" , from \"8a%2B2b=9\" we get
\n" ); document.write( "\"system%28a%2Bb%2Bc=9%2C3a%2Bb=3%2C2a=3%29\"-->\"system%28a%2Bb%2Bc=9%2C3%2A1.5%2Bb=3%2Ca=1.5%29\"-->\"system%28a%2Bb%2Bc=9%2C4.5%2Bb=3%2Ca=1.5%29\"-->\"system%28a%2Bb%2Bc=9%2Cb=-1.5%2Ca=1.5%29\"-->\"system%28a=9%2Cb=-1.5%2Ca=1.5%29\"
\n" ); document.write( "The formula is \"highlight%28x%5Bn%5D=1.5n%5E3-1.5n%5E2%2B9n%2B1%29\"
\n" ); document.write( "We already know that it will give us
\n" ); document.write( "\"x%5B0%5D=1\" , \"x%5B1%5D=10\" , \"x%5B2%5D=25\" , and \"x%5B3%5D=55\" ,
\n" ); document.write( "because we used those terms to find the coefficients.
\n" ); document.write( "Using the formula found for the next given terms, we see that it works:
\n" ); document.write( "\"x%5B4%5D=1.5%2A4%5E3-1.5%2A4%5E2%2B9%2A4%2B1=1.5%2A64-1.5%2A16%2B36%2B1=96-24%2B36%2B1=109\"
\n" ); document.write( "
\n" ); document.write( "\"x%5B6%5D=1.5%2A6%5E3-1.5%2A6%5E2%2B9%2A6%2B1=1.5%2A216-1.5%2A36%2B54%2B1=324-54%2B54%2B1=325\" \n" ); document.write( "
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