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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.
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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