Prove the following by using the principle of mathematical induction for all n ∈ N:

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Solution

Let the given statement be P(n), i.e.,

P(n):

For n = 1, we have

P(1): 1³ = 1 =, which is true.

Let P(k) be true for some positive integer k, i.e.,

We shall now prove that P(k + 1) is true.

Consider

1³ + 2³ + 3³ + … + k³ + (k + 1)³

= (1³ + 2³ + 3³ + …. + k³) + (k + 1)³

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.

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