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kathrynag
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Homework Statement
[tex](1+x)^{k}[/tex][tex]\geq[/tex]1+kx
Homework Equations
The Attempt at a Solution
I want to show for P(k+1)
(1+x)^(k+1)[tex]\geq[/tex]1+kx+x
(1+x)^k*(1+x)[tex]\geq[/tex]1+kx+x
kathrynag said:(1+x)^k*(1+x)[tex]\geq[/tex]1+kx+x
[tex](1+x)^{k}[/tex][tex]\geq[/tex]1+kx
I believe x has to be strictly within 1 unit of 1; i.e., |1 + x| < 1, which means that 0 < x < 2.kathrynag said:So, we have to assume 1+x>0
kathrynag said:No restrictions.
1+kx+x+kx^2
kathrynag said:Once I get here I'm unsure where to go
"Proof by Induction" is a mathematical technique used to prove that a statement or proposition is true for all natural numbers. It involves showing that the statement is true for a starting value, often denoted as P(1), and then proving that if the statement is true for any value k, it must also be true for k+1. This process is repeated until it can be shown that the statement is true for all natural numbers.
"Proof by Induction" is a useful technique because it allows us to prove that a statement is true for infinitely many values without having to check each individual case. It also provides a clear and systematic way to approach proofs involving natural numbers.
The inductive hypothesis in "Proof by Induction" is the assumption that the statement is true for some arbitrary value k. This assumption is used in the proof to show that the statement is also true for the next value, k+1. The inductive hypothesis is crucial in the reasoning behind "Proof by Induction".
The base case in "Proof by Induction" is the starting value for which we show that the statement is true. This is often denoted as P(1) and serves as the foundation for the rest of the proof. It is important to choose a base case that is simple and easy to prove in order to ensure the validity of the overall proof.
No, "Proof by Induction" is specific to natural numbers. It relies on the fact that natural numbers have a well-defined starting point (1) and that there is always a next natural number (k+1). This technique cannot be applied to other types of numbers that do not have these properties, such as real numbers or negative integers.