Mathematical Induction
Математическая Индукция
Induction VS Deduction
Induction – reasoning from the
specific to the general
Deduction – reasoning from the
general to specific
Example:
Example:
I loaned my friend $50 last
November and he failed to pay me
back. (Premise) I loaned him
another $50 just before Christmas,
which he hasn't paid back (Premise),
and yet another $25 in January,
which is still unpaid. (Premise) I
suppose it's time to face facts: He's
never going to pay me back.
(Conclusion)
Everything made of copper conducts
electricity. (Premise)
This wire is made of copper.
(Premise)
This wire will conduct electricity.
(Conclusion)
Source: http://grammar.about.com/od/il/g/inductionterm.htm
Deduction or Induction?
https://www.youtube.com/watch?v=sWia0okDNL8
https://www.youtube.com/watch?v=WVz1-XQIcN8
Source: http://math.stackexchange.com/questions/145189/examples-of-mathematical-induction
Mathematical Induction
How does it work?
Example: prove that 3𝑛 − 1 is a multiple of 2
Step 1: show it is true for n=1
Step 2: assume it is true for n=k
Step 3: prove that 3𝑘+1 − 1 is a multiple of 2
1. Prove that 1 + 2 + 3 + … + n =
𝑛(𝑛+1)
2
2. 𝑛3 + 5𝑛 is a multiple of 6
3. If 0 < 𝑎 < 𝑏, 𝑡ℎ𝑒𝑛 𝑎𝑛 < 𝑏𝑛 (𝑛 ∈ 𝑁)
4. Prove Bernouilli’s Inequality which states:
𝐼𝑓 𝑥 ≥ −1 𝑡ℎ𝑒𝑛 1 + 𝑥
𝑛
≥ 1 + 𝑛𝑥 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑛.