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Find the value of positive integer nnn for which the quadratic equation ∑k=1n(x+k−1)(x+k)=10n\displaystyle \sum_{k = 1}^n(x + k - 1)(x + k) = 10nk=1∑n(x+k−1)(x+k)=10n
has solutions α\alphaα and α+1\alpha + 1α+1 for some value of α\alphaα.
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