I'm not sure what you're saying. Can you please clarify?

In summary, the conversation is about a person seeking help in proving the statement (n + 2)^n ≤ (n + 1)^n+1 where n is a positive integer. They are struggling with the inductive step and are looking for guidance. They mention the base case is easy to solve but struggle with the leap from n=k to n=K+1. The conversation also touches on using parentheses and proper formatting in mathematical expressions.
  • #1
John112
19
0
I need a bit of help proving the following statement
(n + 2)^n ≤ (n + 1)^n+1 where n is a positive integer. The (n+2) and (n+1) bases are making it hard for me solve this. I tried several time, I can't get the inductive step. Can someone lend me a little hand here?

The base case is real simple with n=1; But I can't make the leap from n=k to to n=K+1.
 
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  • #2
you proved it for 1
suppose that (n + 2)^n ≤ (n + 1)^n+1
and prove that ((n+1)+2)^n+1≤ ((n+1)+1)^(n+1)+1
i remember when i first learned about induction these cases kinda confused me as well just go back to the lesson and re read it if you don't get it
 
  • #3
John112 said:
I need a bit of help proving the following statement
(n + 2)^n ≤ (n + 1)^n+1 where n is a positive integer. The (n+2) and (n+1) bases are making it hard for me solve this. I tried several time, I can't get the inductive step. Can someone lend me a little hand here?

The base case is real simple with n=1; But I can't make the leap from n=k to to n=K+1.
Where's the template? PF rules require that you use the template when you post a problem.

Andrax said:
you proved it for 1
suppose that (n + 2)^n ≤ (n + 1)^n+1
and prove that ((n+1)+2)^n+1≤ ((n+1)+1)^(n+1)+1
i remember when i first learned about induction these cases kinda confused me as well just go back to the lesson and re read it if you don't get it

John112 and Andrax - use parentheses!

(n + 1)^n+1 means (n + 1)n + 1

If you intend this as (n + 1)n + 1 then use LaTeX or the SUP tags or at the least, write it as (n + 1)^(n + 1).
 

1. What is an inequality inductive proof?

An inequality inductive proof is a mathematical method used to prove that an inequality is true for all possible values of a variable. It involves using logical reasoning and mathematical operations to establish that the inequality holds for the base case and then showing that it also holds for the next case, and so on until it is proven true for all possible values.

2. How is an inequality inductive proof different from other types of proofs?

Unlike other types of proofs, such as direct and indirect proofs, an inequality inductive proof relies on mathematical induction to establish the truth of an inequality. This means that instead of proving that the inequality is true directly or by contradiction, it is shown to be true for a base case and then for the next case, and so on until it is proven true for all possible values.

3. What are the steps involved in an inequality inductive proof?

The steps involved in an inequality inductive proof are: 1) Establishing the base case, which is usually the smallest or simplest value of the variable in question. 2) Assuming that the inequality holds for a particular value of the variable (this is known as the inductive hypothesis). 3) Using mathematical operations and logical reasoning to show that the inequality also holds for the next value of the variable. 4) Repeating this process until the inequality is proven true for all possible values of the variable.

4. What are some common mistakes to avoid when using an inequality inductive proof?

Some common mistakes to avoid when using an inequality inductive proof are: 1) Not establishing a clear base case. 2) Making incorrect assumptions about the inductive hypothesis. 3) Using incorrect mathematical operations or logical reasoning. 4) Skipping steps in the proof. 5) Failing to consider all possible values of the variable.

5. In what situations is an inequality inductive proof useful?

An inequality inductive proof is useful in situations where it is necessary to prove that an inequality is true for all possible values of a variable. This can be applied in various fields of science, such as physics, economics, and computer science, to prove the validity of mathematical models and theories. It is also commonly used in mathematical competitions and problem-solving exercises.

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