# Equal coefficients

\begin{align} (1008x + 1009)^2 &= 1,016,064x^2 + 2,034,144x + 1,018,081 \\ (1008x + 1009)^3 &= 1,024,192,512x^3 + 3,075,625,728x^2 + 3,078,676,944x + 1,027,243,729 \\ &\vdots \end{align}

What is the smallest positive integer $$n$$ for which the expansion of $$(1008x + 1009)^n$$ has two successive coefficients that are equal?

Details and Assumptions

• As an explicit example, for $$(3x+1)^n$$ the answer is $$n = 3$$ because $$(3x+1)^3 = \underbrace{27x^3 + 27x^2} + 9x + 1$$ has the same coefficient $$27$$ for both $$x^3$$ and $$x^2.$$

• We arrange the terms of the binomial expansion in descending powers of $$x$$.

 Inspiration

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