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Question
Find n, if (n + 3)! = 110 × (n + 1)!
Solution
(n + 3)! = 110 × (n + 1)!
∴ (n + 3)(n + 2)(n + 1)! = 110 × (n + 1)!
∴ (n + 3)(n + 2) = 110
∴ n2 + 5n + 6 = 110
∴ n2 + 5n + 6 − 110 = 0
∴ n2 + 5n − 104 = 0
∴ n2 + 13n − 8n − 104 = 0
∴ n(n + 13) − 8(n + 13) = 0
∴ (n + 13)(n − 8) = 0
∴ n + 13 = 0 or n − 8 = 0
∴ n = − 13 or n = 8
But n ∈ N
∴ n ≠ − 13
Hence, n = 8.
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