You have a sign error. If you can't find it, try expandingmikky05v said:Homework Statement
I have been working on this proof for a few hours and I can not make it work out.
$$\sum_{i=1}^{n}\frac{1}{i(i+1)}=1-\frac{1}{(n+1)}$$
i need to get to $$1-\frac{1}{k+2}$$
I get as far as $$1-\frac{1}{k+1}+\frac{1}{(k+1)(k+2)}$$
then I have tried $$1-\frac{(k+2)+1}{(k+1)(k+2)}$$ bu multiplying the left fraction by (k+2) which got me nowhere.
What am I doing wrong?
Proof by induction is a mathematical technique used to prove that a statement is true for all positive integers or natural numbers. It involves proving that the statement is true for the first or base case, and then showing that if it holds for a particular number, it also holds for the next number, thus creating a chain of logical reasoning that proves the statement is true for all numbers.
Proof by induction can be used to prove statements involving fractions, where the statement is true for all positive rational numbers. This involves proving the statement for the base case, which is usually a simple fraction such as 1/2 or 1/3, and then using the inductive step to show that if the statement holds for a particular fraction, it also holds for the next fraction. This process is repeated until the statement is proven to be true for all positive rational numbers.
The inductive step is the process of showing that if a statement is true for a particular number, it is also true for the next number. In proof by induction, this is done by assuming that the statement is true for a particular number, and then using this assumption to prove that it is also true for the next number. This creates a chain of logical reasoning that proves the statement is true for all numbers.
The base case in proof by induction is the starting point of the proof. It is usually the simplest case, often the number 1, and is used to prove that the statement is true for this particular number. The base case is important because it establishes a foundation for the inductive step, which relies on proving that the statement is true for the next number based on its truth for the previous number.
Some examples of fractions that can be proven using proof by induction include statements about the sum or product of fractions, inequalities involving fractions, and statements about the divisibility of fractions. For example, it can be proven by induction that the sum of any two fractions is also a fraction, or that a fraction multiplied by a positive integer is still a fraction. Other examples could include proving that a certain fraction is always less than another fraction, or that a fraction is divisible by another fraction under certain conditions.