n3 – 7n + 3 is divisible by 3,
Question:

n3 – 7n + 3 is divisible by 3, for all natural numbers n.

Solution:

According to the question,

P(n) = n3 – 7n + 3 is divisible by 3.

So, substituting different values for n, we get,

P(0) = 03 – 7×0 + 3 = 3 which is divisible by 3.

P(1) = 13 – 7×1 + 3 = −3 which is divisible by 3.

P(2) = 23 – 7×2 + 3 = −3 which is divisible by 3.

P(3) = 33 – 7×3 + 3 = 9 which is divisible by 3.

Let P(k) = k3 – 7k + 3 be divisible by 3

So, we get,

⇒ k3 – 7k + 3 = 3x.

Now, we also get that,

⇒  P(k+1) = (k+1)3 – 7(k+1) + 3

= k3 + 3k2 + 3k + 1 – 7k – 7 + 3

= 3x + 3(k2 + k – 2) is divisible by 3.

⇒ P(k+1) is true when P(k) is true.

Therefore, by Mathematical Induction,

P(n) = n3 – 7n + 3 is divisible by 3, for all natural numbers n.

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