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n3 + 5n is divisible by which of the following?

n3 + 5n is divisible by which of the following? Correct Answer 3

P(n) = n3 + 5n P(1) = 1 + 5 P(1) = 6 We assume the P(k) is true and divisible by 6. P(k) = k3 + 5k is divisible by 6 and can be written as 6c or 3 x 2c We need to prove that P(k + 1) is divisible by 6 P(k + 1) = (k + 1)3 + 5(k + 1) P(k + 1) = k3 + 1 + 3k2 + 3k + 5k + 5 P(k + 1) = (k3 + 5k) + 3k2 + 3k + 6 P(k + 1) = 6c + 3(k2 + k + 2) P(k + 1) = (3 x 2c) + 3(k2 + k + 2) Therefore, P(k + 1) is definitely divisible by 3

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