Math Induction: Where Does the >2xk Come From?

In summary, the conversation discusses a proposition involving the inequality 2^(k+1) > k+1 and how it is true for cases where k > 1. The image attached is referred to for further explanation.
  • #1
coconut62
161
1
Please refer to the image attached.

Where does the > 2 x k come from?

Based on the proposition, shouldn't it be > k+1?
 

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  • #2
look two lines above the questioned statement.
 
  • #3
I got it, thanks.
 
  • #4
But one more thing: 2^(k+1) is equal or bigger than k+1. Then it's not necessarily bigger than k+1. How can we say that the proposition is true for that case?
 
  • #5
coconut62 said:
But one more thing: 2^(k+1) is equal or bigger than k+1. Then it's not necessarily bigger than k+1. How can we say that the proposition is true for that case?
If k > 1, then 2k > k + 1.

2k = k + 1 only if k = 1.
 
  • #6
Greater and Greater-Equal

coconut62 said:
But one more thing: 2^(k+1) is equal or bigger than k+1. Then it's not necessarily bigger than k+1. How can we say that the proposition is true for that case?


The propisition is true since it says [itex]2^{k+1} > k + k \ge k+1[/itex] and together [itex]2^{k+1} > k+1[/itex].
 

1. What is mathematical induction?

Mathematical induction is a proof technique used to prove that a statement is true for all natural numbers (usually starting at 0 or 1). It is based on the principle that if a statement is true for one number, and if it is also true for the next number, then it is true for all numbers in between.

2. How does mathematical induction work?

Mathematical induction works by breaking down a statement into smaller cases and proving that each case is true. First, the statement is shown to be true for the base case (usually 0 or 1). Then, assuming the statement is true for some number k, it is proven to be true for the next number, k+1. This creates a chain of logic that proves the statement is true for all numbers.

3. Where does the ">2xk" come from in mathematical induction?

The ">2xk" refers to the general form of the statement being proved in mathematical induction. It is often used when proving statements involving summations or products. In this case, ">2xk" means that the statement is being proved for all numbers greater than or equal to 2 times k.

4. Can mathematical induction be used for any statement?

No, mathematical induction can only be used to prove statements that follow a certain format, known as the principle of mathematical induction. This format involves proving the base case and then showing that if the statement is true for a given number, it is also true for the next number.

5. Are there any limitations to using mathematical induction?

While mathematical induction is a powerful proof technique, it does have its limitations. It can only be used to prove statements about natural numbers, and it may not be the most efficient method for proving certain statements. Additionally, it requires careful attention to detail and can be challenging for some people to understand.

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