- #1
mr_coffee
- 1,629
- 1
Hello everyone.
I've been looking at examples and I can't seem to see what they are doing.
For example, the book has:
Suppose that d1, d2, d3 ... is a sequence defined as follows:
d1 = 9/10, d2 = 10/11.
dk = dk-1 * dk-2 for all inegers k >= 3.
Prove that dn =< 1 for all integers n >= 0.
The book says:
Proof (by strong mathematical induction): let the property P(n) be the inequality dn =< 1.
Show that the proeprty is true for n = 1, and n =2:
I understand why they are starting at n = 1, but why are they testing 2 cases? If i had say, d1 = 9/10, d2 = 10/11, d3 = 10/12, then would i have to test for n = 1, n = 2, and n = 3?
They then go onto say,
Show that for any integer k > 2, if the property is true for all integeers i with 1 =< i < k, then it is true for k:
I'm confused on how they are getting k > 2, also how did they figure out that range from 1 = < i < k ?
My problem looks very similar to this one, but it has 3 subscripts instead of one:
http://suprfile.com/src/1/3itfbah/3.jpg
I would think k >= 3, becuase that's the problem states, that
ek = ek -1, ek-2, ek-3 for all integers k >= 3
THanks!
I've been looking at examples and I can't seem to see what they are doing.
For example, the book has:
Suppose that d1, d2, d3 ... is a sequence defined as follows:
d1 = 9/10, d2 = 10/11.
dk = dk-1 * dk-2 for all inegers k >= 3.
Prove that dn =< 1 for all integers n >= 0.
The book says:
Proof (by strong mathematical induction): let the property P(n) be the inequality dn =< 1.
Show that the proeprty is true for n = 1, and n =2:
I understand why they are starting at n = 1, but why are they testing 2 cases? If i had say, d1 = 9/10, d2 = 10/11, d3 = 10/12, then would i have to test for n = 1, n = 2, and n = 3?
They then go onto say,
Show that for any integer k > 2, if the property is true for all integeers i with 1 =< i < k, then it is true for k:
I'm confused on how they are getting k > 2, also how did they figure out that range from 1 = < i < k ?
My problem looks very similar to this one, but it has 3 subscripts instead of one:
http://suprfile.com/src/1/3itfbah/3.jpg
I would think k >= 3, becuase that's the problem states, that
ek = ek -1, ek-2, ek-3 for all integers k >= 3
THanks!
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