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A sum of products like x(x + 2) + (x + 1)(x + 3) + ... has two integer roots two apart: find n

The same question, asked 2 times: 21 Jan 2026 S2*, 2 Apr 2026 S1. Every wording below, oldest first.

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  1. Question 1

    21 Jan 2026 S2

    51
    The positive integer n , for which the solutions of the equation x(x+2)+(x+2)(x+4)+….+(x+2n−2)(x+2n)=8n3x(x + 2) + (x + 2)(x + 4) + \ldots. +\left( x + 2n - 2 \right)\left( x + 2n \right)=\frac{8n}{3}are two consecutive even integers, is :-

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  2. Question 2

    2 Apr 2026 S1

    51
    Let α,α+2,α∈Z\alpha,\alpha+ 2,\alpha\in Z, be the roots of the quadratic equation x(x+2)+(x+1)(x+3)+(x+2)(x+4)+...+(x+n−1)(x+n+1)=4nx(x+ 2) + (x+ 1)(x+ 3) + (x+ 2)(x+ 4) + ... + (x+n- 1)(x+n+ 1) = 4n for some n∈Nn\in N. Then ( n+αn +\alpha ) is equal to :

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