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function notation. For example, if P (n) is the sentence (formula) ‘n2 + 1 = 3’, then P (1) would
be ‘12 + 1 = 3’, which is false. The construction P (k + 1) would be ‘(k + 1)2 + 1 = 3’. As usual,
this new concept is best illustrated with an example. Returning to our quest to prove the formula
for an arithmetic sequence, we ﬁrst identify P (n) as the formula an = a + (n − 1)d. To prove this
formula is valid for all natural numbers n, we need to do two things. First, we need to establish
that P (1) is true. In other words, is it true that a1 = a + (1 − 1)d? The answer is yes, since this
simpliﬁes to a1 = a, which is part of the deﬁnition of the arithmetic sequence. The second thing
we need to show is that whenever P (k ) is true, it follows that P (k + 1) is true. In other words, we
assume P (k ) is true (this is called the ‘induction hypothesis’) and deduce that P (k + 1) is also
true. Assuming P (k ) to be true seems to invite disaster - after all, isn’t...
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