If a 1 , a 2 , a 3 , ⋯ , a n is an arithmetic sequence where n is a perfect square and n > 4 , then
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ a 1 a n + 1 ⋮ a n − n + 1 a 2 a n + 2 ⋮ a n − n + 2 ⋯ ⋯ ⋯ a n a 2 n ⋮ a n ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ = ?
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Determinants are invariant to adding a multiple of one row to another.
Notice that any row minus the first row is a multiple of the vector of all ones: (1 1 1 ... 1 1 1). This is true because the row elements are consecutive members of the same arithmatic sequence.
Thus, any three rows are colinear, and so the determinant must be 0.
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