4. Prove that for every n > 1,3 divides n³ – n.
Steps to perform Mathematical Induction:
1) Consider an initial value (say n = 1) for which the statement is true, then show that the statement is true for n = initial value.
2) Assume the statement is true for any value of n = k.
3) Then prove the statement is true for n = k+1 by breaking n = k+1 into two parts, one part is n = k (which is already proved) and try to prove the other part.
The given statement is “n3 – n is divisible by 3 for all n ≥ 1”.
The main objective is to prove that for every n ≥ 1, 3 divides n3 – n.
This can be proved by mathematical induction.
For n = 1, (1)3 – 1 = 0 which is divisible by 3.
Thus, the given statement is true for n = 1.
Assume that the statement is true for n = k.
Then, k3 – k is divisible by 3.
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