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Question
By the principle of mathematical induction, prove the following:
52n – 1 is divisible by 24, for all n ∈ N.
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Solution
Let P(n) be the proposition that 52n – 1 is divisible by 24.
For n = 1, P(1) is: 52 – 1 = 25 – 1 = 24, 24 is divisible by 24.
Assume that P(k) is true.
i.e., 52k – 1 is divisible by 24
Let 52k – 1 = 24m
To prove P(k + 1) is true.
i.e., to prove `5^(2(k+1)) - 1` is divisible by 24.
P(k): 52k – 1 is divisible by 24.
P(k + 1) = `5^(2(k+1)) - 1`
= 52k . 52 – 1
= 52k (25) – 1
= 52k (24 + 1) – 1
= 24 . 52k + 52k – 1
= 24 . 52k + 24m
= 24 [52k + 24]
which is divisible by 24 ⇒ P(k + 1) is also true.
Hence by mathematical induction, P(n) is true for all values n ∈ N.
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