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Question
Prove the statement by using the Principle of Mathematical Induction:
n3 – 7n + 3 is divisible by 3, for all natural numbers n.
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Solution
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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