# Proof by induction - fractions

• mikky05v
In summary, proof by induction is a mathematical method used to prove statements for all natural numbers. It can be applied to fractions by treating them as a sequence of rational numbers and using a base case and induction step. This method can be used for all types of fractions and has the advantage of being rigorous and elegant. However, it is limited to proving statements for natural numbers and may not always be the most suitable proof method.
mikky05v

## 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
$$1-\frac{(k+2)+1}{(k+1)(k+2)}$$
by multiplying the left fraction by (k+2) which got me nowhere.

What am I doing wrong?

disregard, I figured it out. I simply had to separate 1/i(i+1) into 1/i - 1/(i+1)

## 1. What is proof by induction?

Proof by induction is a mathematical method used to prove that a statement is true for all natural numbers. It involves proving that the statement is true for a base case, typically when n = 1, and then showing that if the statement is true for n = k, then it must also be true for n = k + 1. This process is repeated until the statement has been proven for all natural numbers.

## 2. How is proof by induction used for fractions?

Proof by induction can be used to prove statements involving fractions by treating the fractions as a sequence of rational numbers. The base case would typically be when the fraction is in its simplest form, such as 1/2. The induction step would involve showing that if the statement is true for a given fraction, then it must also be true for that fraction multiplied by a natural number.

## 3. Can proof by induction be used for all types of fractions?

Proof by induction can be used for all types of fractions as long as they can be expressed as a sequence of rational numbers. This includes proper fractions, improper fractions, mixed numbers, and repeating decimals.

## 4. What are the advantages of using proof by induction for fractions?

Proof by induction is advantageous for fractions because it is a rigorous and systematic method of proving a statement to be true for all natural numbers. It also allows for a concise and elegant proof, as the induction step can be repeated for any natural number, rather than having to prove the statement for each individual case.

## 5. Are there any limitations to using proof by induction for fractions?

One limitation of using proof by induction for fractions is that it can only be used to prove statements for natural numbers. It cannot be used for real or complex numbers. Additionally, some statements involving fractions may be more easily proven using other methods, so it is important to consider the most appropriate proof method for a given statement.

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