Verify that
1(1!)+2(2!)+...+n(n!) = (n+1)! - 1
is true using induction
For the nth case I plug in n = 1
1(1!) = (1+1)! - 1
1 = 2! - 1
1 = 1 This is true
For the n+1 case n = 1
1(1!) + 2(2!)+...+n(n!)(n+1) = ((n+1)+1))! - 1
n(n!)(n+1)((n+1)! - 1) = (n+2)! - 1
1(1!)(1+1)((1+1)!-1) = (1+2)! - 1
2 = 5 This is not true
Therefore, this is not true on the n+1 case.
Am I correct?
The first line should probably be
1(1!) + 2(2!)+...+(n+1)(n!)(n+1) = ((n+1)+1))! - 1
Also, I don't understand how the second line is related to the first one.
In general, proofs in the high school style consisting of a series of equalities ending with something like 1 = 1 are not used in higher mathematics. In a proof, it should be clear how subsequent claims follow from previous ones. Thus, real proofs may use long chains of equalities E1 = E2 = E3 = ..., possibly split over several lines and accompanied by short comments saying how each equality is obtained. Otherwise, a proof should be a sequence of claims joined by words or symbols such as "therefore", "thus" or "⇒", which show the relationship between the claims.
Now I am not sure what you mean by n-case and (n+1)-case. A proof by induction starts with identifying a property of natural numbers, often denoted by P(n). For each n, P(n) is either true or false (in particular, P(n) is not a number). This property can be an equality, as in the current problem, but it can have other forms: e.g., e_{1}(n) divides e_{2}(n) for some expressions e_{1} and e_{2}, e(n) is a triangular number for some e, etc. The problem is to prove that P(n) holds for all natural n ≥ n_{0}. The base case consists of proving P(n_{0}). The induction step consists of proving that P(n) implies P(n+1) for all n ≥ n_{0}.
In this problem, P(n) is 1(1!) + 2(2!) + ... + n(n!) = (n+1)! - 1; therefore, P(n+1) is 1(1!) + 2(2!) + ... + (n+1)(n+1)! = (n+2)! - 1. I assumed the line
1(1!) + 2(2!)+...+n(n!)(n+1) = ((n+1)+1))! - 1
in post #1 supposed to be P(n+1). That's why I said n(n!)(n+1) should be replaced by (n+1)(n!)(n+1) = (n+1)(n+1)!.
[QUOTE=RatchetTheLombax;785496]You are probably right. Induction doesn't make any sense at all. I
understand what it is but that is the extent of it.[/QUOTE
Why do you say that?
Here is a real-world example.
Think of a garden path made up of stepping stones.
Now suppose I show you that there is absolutely a way that you can get onto the first stone in the path.
Then I show you that if you are on any stone then you can absolutely 'move' to the next stone.
Does that not mean that "you can start at the first stone and move through the entire garden path?
You can get on at , but then you have a grantee you can go to then to , etc.
That is what induction is all about.
We can step on all positive integers for a statement.