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Question
Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:
1 + 3 + 5 + ... + (2n – 1) = n2
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Solution
Let the given statement P(n) be defined as P(n) : 1 + 3 + 5 + ... + (2n – 1) = n2, for n ∈ N.
Note that P(1) is true
Since P(1) : 1 = 12
Assume that P(k) is true for some k ∈ N
i.e., P(k) : 1 + 3 + 5 + ... + (2k – 1) = k2
Now, to prove that P(k + 1) is true
We have 1 + 3 + 5 + ... + (2k – 1) + (2k + 1)
= k + (2k + 1)
= k2 + 2k + 1
= (k + 1)2
Thus, P(k + 1) is true, whenever P(k) is true.
Hence, by the Principle of Mathematical Induction, P(n) is true for all n ∈ N.
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