- #1
Mark44 said:For starters, you have a typo in your summation. It should be
[tex]\sum_{o = 1}^n i\cdot i![/tex]
The index for your summation is i, not o or 0.
Your induction hypothesis is
[tex]\sum_{i = 1}^k i\cdot i! = (k + 1)! - 1[/tex]
This is the statement when n = k
The statement you're trying to prove, when n = k + 1, is
[tex]\sum_{i = 1}^{k + 1} i\cdot i! = (k + 2)! - 1[/tex]
and not what you have.
Mathematical induction is a method of mathematical proof used to prove that a statement is true for all natural numbers (positive integers). It involves proving a base case and then using a logical argument to show that if the statement is true for a particular number, it is also true for the next number.
Mathematical induction works by proving the statement for a base case, usually the number 1 or 0. Then, using the fact that the statement is true for this base case, it is proven that the statement is also true for the next number in the sequence. This process is repeated until the statement is proven to be true for all natural numbers.
Mathematical induction is used when trying to prove a statement that is true for all natural numbers. It is commonly used in algebra, number theory, and discrete mathematics.
No, mathematical induction can only be used to prove statements that are true for all natural numbers. It cannot be used to prove statements that are only true for a finite number of cases or for real numbers.
The key steps in a mathematical induction proof include proving the base case, assuming the statement is true for a particular number, using this assumption to prove the statement for the next number, and then repeating this process until the statement is proven to be true for all natural numbers.