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Question
Prove the following by using the principle of mathematical induction for all n ∈ N:
1.2 + 2.3 + 3.4+ ... + n(n+1) = `[(n(n+1)(n+2))/3]`
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Solution
Let the given statement be P(n), i.e.,
We shall now prove that P(k + 1) is true.
Consider
1.2 + 2.3 + 3.4 + … + k.(k + 1) + (k + 1).(k + 2)
= [1.2 + 2.3 + 3.4 + … + k.(k + 1)] + (k + 1).(k + 2)
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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