Mathematical induction question

In summary, mathematical induction is a proof technique that uses recursion to prove statements about mathematical objects with a defined structure. It differs from other proof techniques because it relies on the principle of recursion and involves proving a base case and then using an inductive hypothesis to prove the next case. The steps for using mathematical induction are to prove the base case, assume the inductive hypothesis, use it to prove the next case, and repeat until all cases are proven. This technique can be used for any mathematical object with a defined structure, and it is useful for proving statements about sequences, series, and recursive structures. Some common mistakes to avoid when using mathematical induction include assuming the statement is true without proof and using the wrong base case.
  • #1
Googl
111
1
Hi all,

I am revising on Proof by mathematical induction and I have came across a question I haven't found a way to work it out.

[tex]\sum_{r=1}^n r(r!) = (n + 1)! -1[/tex]

I understand the steps of proving by mathematical induction question. The ! is causing the confusion.
 
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  • #3
Maybe you can rewrite the general summand as:

(r+1)r!-r! and expand the sum, see what happens.
 

FAQ: Mathematical induction question

What is mathematical induction?

Mathematical induction is a proof technique used to prove statements about mathematical objects that have a defined structure, such as natural numbers or mathematical functions. It involves proving that a statement holds for a base case, and then showing that if it holds for a certain value, it also holds for the next value.

How does mathematical induction differ from other proof techniques?

Mathematical induction is different from other proof techniques, such as direct proof or proof by contradiction, because it relies on the principle of recursion. This means that the proof for a particular value relies on the proof for the previous value, and this chain of reasoning continues until the base case is reached.

What are the steps for using mathematical induction?

The steps for using mathematical induction are as follows:1. Start by proving the base case, which is usually the smallest or simplest value of the mathematical object.2. Assume that the statement holds for a certain value, called the "inductive hypothesis".3. Use the inductive hypothesis to prove that the statement also holds for the next value.4. Repeat this process until the statement has been proven for all values.

What types of statements can be proven using mathematical induction?

Mathematical induction can be used to prove statements about any mathematical object that has a defined structure, such as natural numbers, mathematical functions, or even geometric figures. It is particularly useful for proving statements about sequences, series, and other recursive structures.

Are there any common mistakes to avoid when using mathematical induction?

One common mistake to avoid when using mathematical induction is assuming that the statement holds for all values without actually proving it. It is important to carefully follow the steps of mathematical induction and ensure that the statement is proven for each value. Another mistake is using the wrong base case, which can lead to an incorrect proof. It is also important to correctly apply the inductive hypothesis and use it to prove the statement for the next value, rather than assuming it holds without proof.

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