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Question
Prove the statement by using the Principle of Mathematical Induction:
32n – 1 is divisible by 8, for all natural numbers n.
Solution
P(n) = 32n – 1 is divisible by 8.
So, substituting different values for n, we get,
P(0) = 30 – 1 = 0 which is divisible by 8.
P(1) = 32 – 1 = 8 which is divisible by 8.
P(2) = 34 – 1 = 80 which is divisible by 8.
P(3) = 36 – 1 = 728 which is divisible by 8.
Let P(k) = 32k – 1 be divisible by 8.
So, we get,
⇒ 32k – 1 = 8x
Now, we also get that,
⇒ P(k + 1) = `3^(2("k" + 1))` – 1
= 32(8x + 1) – 1
= 72x + 8 is divisible by 8.
⇒ P(k + 1) is true when P(k) is true.
Therefore, by Mathematical Induction, P(n) = 32n – 1 is divisible by 8, for all natural numbers n.
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