I was wondering if it is possible to find the exact solution to the following type of polynomial. \begin{equation*} x+x^2+x^3+x^4+...x^n=const \end{equation*} Where the coefficient for each polynomial term is always 1. It is very similar to the 'all one polynomial' except the constant is not 1. I know that not all polynomials have an analytical solution, but I was wondering if a special case such as this might.
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3It may be worthwhile noting that if $x \neq 1$, then $x + x^2 + \cdots + x^n = x(1 + x + \cdots + x^{n-1}) = x \cdot \frac{x^n - 1}{x-1}$, so we are trying to solve $x^{n+1} - x = C(x-1)$, or $x^{n+1} - (C+1)x + C = 0$. – user217285 Mar 14 '16 at 22:05
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You can find it by recursion, but it will not be a closed formula. Similar to what I have done here: http://www.genautica.com/math/naccis/fibonacci_and_higher_order_naccis.html – Stefan Gruenwald Aug 01 '16 at 21:01
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If possible I was trying to avoid recursion or iteration based solutions for the purpose of optimizing computational run time. This is a procedure that will be performed many, many times with different constants each time – Travis Aug 03 '16 at 22:23