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Only for natural numbers.etotheipi said:Is it only for natural numbers? Why do you need induction, can't you just multiply out the bracket?
You read the problem right. Thanks for the answer.Mark44 said:What is the actual problem statement? The image you uploaded is almost impossible to read. As far as I can tell, it looks like you are to prove that ##2n + 7 > (n + 3)^2##.
You don't need induction to prove this. You can do a proof by contradiction. I.e., assume that ##2n + 7 \le (n + 3)
^2##. Expand the right side and from there get a contradiction.
Well, lucky for me. The image you posted was nearly unreadable, which is why we ask that members posting homework problems type the equations or inequalities rather than post an image of their work. It's very rare that someone posts an image that can easily be read. Most images are unreadable due to illegible writing or poor lighting of the work being photographed, or a combination of the two.sahilmm15 said:You read the problem right.
PMI is a mathematical proof technique that is used to prove a statement or formula for all natural numbers. It involves two main steps: the base case, where the statement is proven to be true for the first natural number, and the inductive step, where it is assumed that the statement is true for some arbitrary natural number and then proven to be true for the next natural number. This process is repeated until the statement is proven to be true for all natural numbers.
The k+1 step is the inductive step in the PMI process. It is where we assume that the statement is true for some arbitrary natural number k and then prove that it is also true for the next natural number, k+1. This step is crucial in proving that the statement holds true for all natural numbers.
PMI is typically used to prove statements or formulas that involve natural numbers, such as equations, inequalities, and divisibility. If a problem involves proving something for all natural numbers, then PMI may be a useful tool to use.
No, PMI is specifically used for proving statements for natural numbers. It cannot be used for other types of numbers, such as real numbers or complex numbers.
One limitation of PMI is that it can only be used to prove statements for natural numbers. Additionally, it may not be the most efficient method for solving some problems, as it requires multiple steps and can be time-consuming. It is also important to ensure that the base case and inductive step are correctly formulated in order to prove the statement correctly.