Proving 2^n via Induction: 1+n!/1!+n(n-1)!/2!+...”

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In summary, the purpose of proving 2^n via induction is to establish the validity of a mathematical statement or formula. Induction is a mathematical proof technique that involves breaking down a larger problem into smaller, more manageable parts and proving a base case and inductive hypothesis. The inductive hypothesis in this proof is that the expression 1+n!/1!+n(n-1)!/2!+... is equal to 2^n for some positive integer k. Proving mathematical statements is important because it allows us to confidently use them in further calculations and understand the underlying principles in mathematics.
  • #1
theperthvan
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How can I show that

[tex]1+\frac{n}{1!}+\frac{n(n-1)}{2!}+\frac{n(n-1)(n-2)}{3!}+...= 2^{n} [/tex]

This comes from proving that the power set of a set with n elements is [tex]2^{n}[/tex].

I got so far that nCn+nC(n-1)+ ... = what I have above. Now for the induction...
Cheers,
 
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  • #2
Hint: What's the binomial expansion of (1 + 1)^n?
 
  • #3
of course, thanks
 

1. What is the purpose of proving 2^n via induction?

The purpose of proving 2^n via induction is to establish the validity of a mathematical statement or formula. In this case, we are trying to prove that the expression 1+n!/1!+n(n-1)!/2!+... is equal to 2^n for all positive integers n.

2. What is induction in mathematics?

Induction is a mathematical proof technique used to prove that a statement is true for all values within a certain range. It involves proving a base case (usually the smallest value in the range) and then showing that if the statement is true for one value, it is also true for the next value in the sequence.

3. How does induction work?

Induction works by breaking down a larger problem into smaller, more manageable parts. The base case is proved to be true, and then the inductive hypothesis is used to show that if the statement is true for one value, it must also be true for the next value. This process continues until the entire range of values has been proven.

4. What is the inductive hypothesis in this proof?

The inductive hypothesis in this proof is that the expression 1+n!/1!+n(n-1)!/2!+... is equal to 2^n for some positive integer k. This is used to prove that the statement is also true for the next value, k+1.

5. Why is it important to prove mathematical statements?

Proving mathematical statements is important because it allows us to establish the validity of a statement or formula, and therefore use it confidently in further calculations and proofs. It also helps us to understand the underlying principles and patterns in mathematics.

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