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Stitz-Zeager_College_Algebra_e-book

# 2 since one revolution is 2 radians and t 2 is

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Unformatted text preview: ls, a, a + d, a + 2d, a + 3d, a + 4d + . . . The pattern suggested here is that to reach the nth term, we start with a and add d to it exactly n − 1 times, which lead us to our formula an = a + (n − 1)d for n ≥ 1. But how do we prove this to be the case? We have the following. The Principle of Mathematical Induction (PMI): Suppose P (n) is a sentence involving the natural number n. IF 1. P (1) is true and 2. whenever P (k ) is true, it follows that P (k + 1) is also true THEN the sentence P (n) is true for all natural numbers n. The Principle of Mathematical Induction, or PMI for short, is exactly that - a principle.1 It is a property of the natural numbers we either choose to accept or reject. In English, it says that if we want to prove that a formula works for all natural numbers n, we start by showing it is true for n = 1 (the ‘base step’) and then show that if it is true for a generic natural number k , it must be true for the next natural number, k + 1 (the ‘inductive step’). The notation P (n) acts just l...
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