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Question
Prove the statement by using the Principle of Mathematical Induction:
4n – 1 is divisible by 3, for each natural number n.
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Solution
P(n) = 4n – 1 is divisible by 3.
So, substituting different values for n, we get,
P(0) = 40 – 1 = 0 which is divisible by 3.
P(1) = 41 – 1 = 3 which is divisible by 3.
P(2) = 42 – 1 = 15 which is divisible by 3.
P(3) = 43 – 1 = 63 which is divisible by 3.
Let P(k) = 4k – 1 be divisible by 3,
So, we get,
⇒ 4k – 1 = 3x.
Now, we also get that,
⇒ P(k + 1) = 4k+1 – 1
= 4(3x + 1) – 1
= 12x + 3 is divisible by 3.
⇒ P(k + 1) is true when P(k) is true
Therefore, by Mathematical Induction,
P(n) = 4n – 1 is divisible by 3 is true for each natural number n.
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