Unformatted text preview: + d = a + kd = a + ((k + 1) − 1)d.
Hence, P (k + 1) is true.
In essence, by showing that P (k + 1) must always be true when P (k ) is true, we are showing that
the formula P (1) can be used to get the formula P (2), which in turn can be used to derive the
formula P (3), which in turn can be used to establish the formula P (4), and so on. Thus as long
as P (k ) is true for some natural number k , P (n) is true for all of the natural numbers n which
follow k . Coupling this with the fact P (1) is true, we have established P (k ) is true for all natural
numbers which follow n = 1, in other words, all natural numbers n. One might liken Mathematical
Induction to a repetitive process like climbing stairs.2 If you are sure that (1) you can get on the
stairs (the base case) and (2) you can climb from any one step to the next step (the inductive step),
then presumably you can climb the entire staircase.3 We get some more practice with induction in
the following example.
Example 9.3.1. Prove the following assertions using the Principle of Mathematical Induction.
n 1. The sum formula for arithmetic sequences: (a...
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