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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