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Question
Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:
2n + 1 < 2n, for all natual numbers n ≥ 3.
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Solution
Let P(n) be the given statement
i.e., P(n) : (2n + 1) < 2n for all natural numbers, n ≥ 3.
We observe that P(3) is true
Since 2.3 + 1 = 7 < 8 = 23
Assume that P(n) is true for some natural number k
i.e., 2k + 1 < 2k
To prove P(k + 1) is true
We have to show that 2(k + 1) + 1 < 2k+1
Now, we have 2(k + 1) + 1 = 2k + 3
= 2k + 1 + 2 < 2k + 2 < 2k . 2
= 2k + 1.
Thus P(k + 1) is true, whenever P(k) is true.
Hence, by the Principle of Mathematical Induction P(n) is true for all natural numbers, n ≥ 3.
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