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I have a particular number series as follows:-7, 3, 5, 13 and 27. Please assume as correct that -7 represent 7 points in 0D, 3 represents 3 points in 1D, 5 points for 2D, 13 points for 3D, and 27 points in 4D. Also assume that -7 will be understood as its absolute value.

First, you have +3-7=-4




Next, you have: +2-(-4)=+6



Therefore, the arithmetic series -7 (Absolute Value), 3, 5, 13, and 27 reduces to a single constant term - 6.

Here is my question: I would like to find out what the above inductive proof indicates, if anything, regarding this particular arithmetic sequence. I understanding that the series reduces to a constant term, but what, if anything, does this mean?

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You are finding that the second differences are constant. That implies that there is a quadratic $an^2+bn+c$ that fits all your data (but with $7$ in place of $-7$).

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