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Statement 3.38: If n is a natural number, then 4 divides 5^n - 1 :lock:,…
Statement 3.38: If n is a natural number, then 4 divides 5^n - 1 :lock:
For each natural number n,
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P(1)
Induction step
Assume that for some natural number k,
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P(k)
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Call this IH
Thus 4|5^k -1
Thus, 4l =5^k-1
For some integer l
5^{k+1} -1
=5(5^k)-1
=5(4l+1) -1
=5(4l)+5-1
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then P(k+1).
We proved
If P(k),
P(k+1)
By the axiom of induction, for every natural number n we have P(n)
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