# Prove by Mathematical Induction 1/1.2+1/2.3+...+1/(n(n+1))=n/(n+1)?

Jun 22, 2017

#### Explanation:

Induction method is used to prove a statement. Most commonly, it is used to prove a statement, involving, say $n$ where $n$ represents the set of all natural numbers.

Induction method involves two steps, One, that the statement is true for $n = 1$ and say $n = 2$. Two, we assume that it is true for $n = k$ and prove that if it is true for $n = k$, then it is also true for $n = k + 1$.

First Step $-$ Now for $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \ldots + \frac{1}{n \left(n + 1\right)} = \frac{n}{n + 1}$, we know for $n = 1$, we have $\frac{1}{1 \cdot 2} = \frac{1}{2}$ and for $n = 2$, we have $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} = \frac{1}{2} + \frac{1}{6} = \frac{2}{3} = \frac{2}{2 + 1}$.

Hence, given statement is true for $n = 1$ and $n = 2$.

Second Step $-$ Assume it is true for $n = k$, hence

$\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \ldots + \frac{1}{k \left(k + 1\right)} = \frac{k}{k + 1}$

Now let us test it for $n = k + 1$ i.e.

$\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \ldots + \frac{1}{k \left(k + 1\right)} + \frac{1}{\left(k + 1\right) \left(k + 2\right)}$

= $\frac{k}{k + 1} + \frac{1}{\left(k + 1\right) \left(k + 2\right)}$

= $\frac{k \left(k + 2\right) + 1}{\left(k + 1\right) \left(k + 2\right)}$

= $\frac{{k}^{2} + 2 k + 1}{\left(k + 1\right) \left(k + 2\right)}$

= ${\left(k + 1\right)}^{2} / \left(\left(k + 1\right) \left(k + 2\right)\right)$

= $\frac{k + 1}{k + 2}$

Hence we see that the statement is true for $n = k + 1$ if it is true for $n = k$.

Hence $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \ldots + \frac{1}{n \left(n + 1\right)} = \frac{n}{n + 1}$