Transcript Induction

Mathematical Induction
• A common proof technique
• Let P be a proposition to prove, and express P
in terms of a positive integer parameter n. To
show that P(n) is true, follow these steps:
1. Base Step: Verify that P(1) is true
2. Induction Hypothesis: Assume that P(n) is
true
for some n  1
3. Inductive Step: Show that P(n + 1) is also
Mathematical Induction
• Note that the inductive step is the only step that
really requires real work. In particular,
– The base step usually only requires simple checking on
what happens when n = 1
– No proof or computation is required for the induction
hypothesis. We simply proclaim the assumption that
P(n) is true
– In the inductive step, we want to prove that P(n + 1) is
true. Typically, this is done by applying the
information gathered from P(n). In other words, we
are trying to show that P(n)  P(n + 1)
Mathematical Induction – Example
• Prove by mathematical induction that
n
f(n) =
i
= n (n + 1)
2
i=1
where n > 0
Example
• Prove by mathematical induction that the
sum of the first n positive odd numbers is
n2.
• Hint: note that the nth odd number is
simply 2n - 1. So, all you have to prove is
the following: