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

, 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

(1.3 + 3.5 + 5.7 + … + (2k – 1) (2k + 1) + {2(k + 1) – 1}{2(k + 1) + 1}

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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n \in N : 1.3 + 3.5 + 5.7 + . . . . . . . + (2 n - 1) (2 n + 1) = \frac{n (4 n^{2} + 6 n - 1)}{3}

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

1.3 + 3.5 + 5.7 + . . . + (2 n - 1) (2 n + 1) = \frac{n (4 n^{2} + 6 n - 1)}{3}

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