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8 |
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12 |
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| 285 |
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255 |
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247 |
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221 |
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209 |
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187 |
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30 |
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26 |
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22 |
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The numbers in red are projections determining what follows 209.
In the above table second row we have the given numbers in sequence. Now we write the differences between the numbers as shown in two different rows.The other rows show differences which differ by 4 (12-8=4, 30-26=4). In the first row the differences are increasing by 4 while in the bottom row they're decreasing by 4. If this is the pattern then we would expect there to be a difference of 4 in the bottom row so 26-4=22 (in red). If we were to continue the top row the next difference would be 12+4=16 and would generate the number following 187, that is 187-16=171. The next two numbers in the sequence would be 187 and 171 following this pattern. The shape of all the numbers in the table forms a zigzag wave.
Another way to figure out the sequence pattern is to use a function:
f(n)=a+bn+cn2+dn3+en4, where integer n, 0≤n≤4, is the term number starting at n=0. We would then have a system of equations which would enable us to calculate the coefficients a-e.
f(0)=a=285; f(1)=a+b+c+d+e=285+b+c+d+e=255; f(2)=285+2b+4c+8d+16e=247; f(3)=285+3b+9c+27d+81e=221; f(4)=285+4b+16c+64d+256e=209. By a process of progressive elimination the values of b, c, d, e can be found. f(n) then can be used to find any term in the sequence, so f(5) would produce the term following 209. If necessary I can show you how to find these coefficients. The actual function is:
f(n)=285-217n/3+64n2-74n3/3+3n4; f(5)=315, a surprise result since the term is not lower than 209.
There are other methods to find a formula linking the terms, but they will most probably produce different formulas and therefore different terms to follow 209.