Hi all,
Awesome site you've got here.
I'm stuck on a problem in an intro to proof/numbers/set theory/functions book I'm working through in my spare time.
Apologies if this is in the wrong forum. This isn't a calculus problem (AFAIK), but I didn't see anywhere more appropriate, so here it is.
Anyway, the question is this:
Given a sequence of numbers \(\displaystyle a(1), a(2),...,\) the number \(\displaystyle \prod_{i=1}^{n} a(i)\) is defined inductively by,
(i) \(\displaystyle \prod_{i=1}^{1}a(i)=a(1)\)
(ii) \(\displaystyle \prod_{i=1}^{k+1}a(i)=(\prod_{i=1}^{k} a(i))\cdot a(k+1)\) for \(\displaystyle k\geq1\)
Prove that \(\displaystyle \prod_{i=1}^{n}(1+x^{2^{i-1}})=(1-x^{2^{n}})/(1-x)\) for \(\displaystyle x\not=1\)
So far, that all looks quite straightforward.
The way the book teaches you to structure inductive proofs is as follows:
Proof: We use induction on \(\displaystyle n\).
Base case: [Prove statement \(\displaystyle P(1)\)]
Inductive step: Suppose as inductive hypothesis that [\(\displaystyle P(k)\) is true] for some positive integer k. Then [deduce that \(\displaystyle P(k+1)\) is true]. This proves the inductive step.
Conclusion: Hence, by induction, [\(\displaystyle P(n)\) is true] for all positive integers \(\displaystyle n\).
Again, all pretty straight forward.
But with this problem, the base case appear to be false!
For \(\displaystyle n=1\), \(\displaystyle (1+x^{2^{0}})=1+x\) and \(\displaystyle (1-x^{2^{0}})/(1-x)=1\).
Consequently, my proof has fallen at the first hurdle.
Can anyone help? Is there an error in the text? Or maybe I'm missing something obvious...?
Cheers
Awesome site you've got here.
I'm stuck on a problem in an intro to proof/numbers/set theory/functions book I'm working through in my spare time.
Apologies if this is in the wrong forum. This isn't a calculus problem (AFAIK), but I didn't see anywhere more appropriate, so here it is.
Anyway, the question is this:
Given a sequence of numbers \(\displaystyle a(1), a(2),...,\) the number \(\displaystyle \prod_{i=1}^{n} a(i)\) is defined inductively by,
(i) \(\displaystyle \prod_{i=1}^{1}a(i)=a(1)\)
(ii) \(\displaystyle \prod_{i=1}^{k+1}a(i)=(\prod_{i=1}^{k} a(i))\cdot a(k+1)\) for \(\displaystyle k\geq1\)
Prove that \(\displaystyle \prod_{i=1}^{n}(1+x^{2^{i-1}})=(1-x^{2^{n}})/(1-x)\) for \(\displaystyle x\not=1\)
So far, that all looks quite straightforward.
The way the book teaches you to structure inductive proofs is as follows:
Proof: We use induction on \(\displaystyle n\).
Base case: [Prove statement \(\displaystyle P(1)\)]
Inductive step: Suppose as inductive hypothesis that [\(\displaystyle P(k)\) is true] for some positive integer k. Then [deduce that \(\displaystyle P(k+1)\) is true]. This proves the inductive step.
Conclusion: Hence, by induction, [\(\displaystyle P(n)\) is true] for all positive integers \(\displaystyle n\).
Again, all pretty straight forward.
But with this problem, the base case appear to be false!
For \(\displaystyle n=1\), \(\displaystyle (1+x^{2^{0}})=1+x\) and \(\displaystyle (1-x^{2^{0}})/(1-x)=1\).
Consequently, my proof has fallen at the first hurdle.
Can anyone help? Is there an error in the text? Or maybe I'm missing something obvious...?
Cheers