- #1
samii
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if a>1 and ⁿ√a = 1 + x, prove that 0 < x < a/n
Deduce that ⁿ√a →1, n →∞
confused!
Deduce that ⁿ√a →1, n →∞
confused!
Proof Deduction: a>1 & ⁿ√a = 1+x | Prove 0 < x < a/n
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In summary, proof deduction is a method used in mathematics and logic to prove the truth of a statement or proposition by starting from a set of given premises and using logical reasoning to arrive at a conclusion. An example of this in action is proving the Pythagorean theorem by manipulating equations and using geometric properties. The key components of proof deduction include given premises, logical reasoning, and arriving at a logical conclusion. It differs from other methods of proof, such as induction or contradiction, as it relies on logical reasoning and equations rather than examples or assumptions.
- #2
tiny-tim
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hi samii! 
rewrite n√a = 1 + x
rewrite n√a = 1 + x
- #3
HallsofIvy
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If [itex]\sqrt[n]{a}= 1+ x[/itex], then [itex]a= (1+ x)^n[/itex]. Expand using the binomial theorem to get that a= 1+ nx+ positive terms so that 1+ nx< a. Since x< a/n, as n goes to infinity, x goes to 0.
- #4
samii
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hii again :)
thanks also if 0 < a < 1
what is the corresponding result?
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tiny-tim
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then n√a would be < 1, so you'd have to write n√a = 1 - x …
so what would happen then?
1. What is proof deduction?
Proof deduction is a method used in mathematics and logic to prove the truth of a statement or proposition by starting from a set of given premises and using logical reasoning to arrive at a conclusion.
2. How do you prove that a>1 and ⁿ√a = 1+x leads to the inequality 0 < x < a/n?
To prove this, we can start by assuming that a>1 and ⁿ√a = 1+x are both true. From the given equation, we can manipulate it to get x alone on one side, which gives us x = ⁿ√a - 1. We can then substitute this value of x into the inequality, which gives us ⁿ√a - 1 > 0 and ⁿ√a - 1 < a/n. By combining these two inequalities, we get 0 < x < a/n, which proves the original statement.
3. Can you provide an example of proof deduction in action?
One example of proof deduction is proving the Pythagorean theorem, where we start with the given premises of a right triangle with sides a, b, and c, and use logical reasoning to arrive at the conclusion that a² + b² = c². This is done by manipulating the algebraic equations and using geometric properties of right triangles.
4. What are the key components of proof deduction?
The key components of proof deduction include starting with given premises, using logical reasoning to manipulate and manipulate equations, and arriving at a logical conclusion that follows from the given premises.
5. How is proof deduction different from other methods of proof?
Proof deduction is different from other methods of proof, such as proof by induction or proof by contradiction, as it relies on logical reasoning and manipulating equations rather than using specific examples or assumptions to prove a statement.
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