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www/statics/euler/Euler_Problem-101_description.md

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+If we are presented with the first k terms of a 
+sequence it is impossible to say with certainty the value of the next term, 
+as there are infinitely many polynomial functions that can model the sequence.
+
+As an example, let us consider the sequence of cube numbers. 
+This is defined by the generating function,
+
+~~~
+u_n = n^3: 1, 8, 27, 64, 125, 216, ...
+~~~
+
+Suppose we were only given the first two terms of this sequence.
+Working on the principle that "simple is best" we should assume a linear 
+relationship and predict the next term to be 15 (common difference 7).
+Even if we were presented with the first three terms, by the same 
+principle of simplicity, a quadratic relationship should be assumed.
+
+We shall define `OP(k, n)` to be the nth term of the optimum polynomial 
+generating function for the first k terms of a sequence.
+It should be clear that `OP(k, n)` will accurately generate the terms 
+of the sequence for `n <= k`, and potentially the first incorrect term (FIT) 
+will be `OP(k, k+1)`; in which case we shall call it a bad OP (BOP).
+
+As a basis, if we were only given the first term of sequence, 
+it would be most sensible to assume constancy; that is, for `n >= 2`, `OP(1, n) = u_1`.
+
+Hence we obtain the following OPs for the cubic sequence:
+
+~~~
+OP(1, n) = 1            1,     **1**, 1,      1,           ...
+OP(2, n) = 7n-6         1,     8,     **15**,              ...
+OP(3, n) = 6n^2-11n+6   1,     8,     27,   **58**,        ...
+OP(4, n) = n^3          1,     8,     27,     64,     125, ...
+~~~
+
+Clearly no BOPs exist for `k >= 4`.
+
+By considering the sum of FITs generated by the BOPs (indicated in **red** above), 
+we obtain `1 + 15 + 58 = 74`.
+
+Consider the following tenth degree polynomial generating function:
+
+~~~
+un = 1 - n + n^2 - n^3 + n^4 - n^5 + n^6 - n^7 + n^8 - n^9 + n^10
+~~~
+
+Find the sum of FITs for the BOPs.