Math 308 Lecture 10 - Mathematical Induction

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[kstahmer]

Suppose n = 1, 2, 3...

Note: 5^-1 ≡ 3 (mod 7) because 5(3) ≡ 1 (mod 7)
Also, 3^4 ≡ 81 ≡ 4 (mod 7) and 5^2 ≡ 25 ≡ 4 (mod 7)

3^(4n + 1) – 5^(2n - 1) ≡ 3(3^(4n)) – 3(5^(2n))

≡ 3(3^(4n) – 5^(2n)) ≡ 3(4^n – 4^n)

≡ 3(0) ≡ 0 (mod 7)

Interpreting 5^-1 as 3 (mod 7) extends n to:

Suppose n is an integer.