Question

Find value of $\dfrac{1}{{1.2}} + \dfrac{1}{{2.3}} + \dfrac{1}{{3.4}} + .... + \dfrac{1}{{n\left( {n + 1} \right)}}$.
A. $\dfrac{1}{{n\left( {n + 1} \right)}}$
B. $\dfrac{n}{{\left( {n + 1} \right)}}$
C. $\dfrac{{2n}}{{\left( {n + 1} \right)}}$
D. $\dfrac{2}{{n\left( {n + 1} \right)}}$

Answer 1

Hint: The value of the expression $\dfrac{1}{{1.2}} + \dfrac{1}{{2.3}} + \dfrac{1}{{3.4}} + .... + \dfrac{1}{{n\left( {n + 1} \right)}}$ can be found easily by splitting each term into two parts. We will first transform the numerator of each term and split the terms into two parts such that it is easy to find the summation of the terms.

Complete step by step solution:
So, we have, $\dfrac{1}{{1.2}} + \dfrac{1}{{2.3}} + \dfrac{1}{{3.4}} + .... + \dfrac{1}{{n\left( {n + 1} \right)}}$
We can clearly see that the denominator of each term is the product of two consecutive numbers. So, we can replace the numerator of each term by the difference of the terms present in the denominator.
So, we get,
$ \Rightarrow \dfrac{{2 - 1}}{{1.2}} + \dfrac{{3 - 2}}{{2.3}} + \dfrac{{4 - 3}}{{3.4}} + .... + \dfrac{{\left( {n + 1} \right) - n}}{{n\left( {n + 1} \right)}}$
We split up each term into two parts.
$ \Rightarrow \left( {\dfrac{1}{1} - \dfrac{1}{2}} \right) + \left( {\dfrac{1}{2} - \dfrac{1}{3}} \right) + \left( {\dfrac{1}{3} - \dfrac{1}{4}} \right) + \left( {\dfrac{1}{4} - \dfrac{1}{5}} \right).... + \left( {\dfrac{1}{n} - \dfrac{1}{{\left( {n + 1} \right)}}} \right)$
The negative term in a bracket has the same magnitude as the positive in the next bracket.
Canceling all the like terms with same magnitude, we get,
$ \Rightarrow 1 - \dfrac{1}{{\left( {n + 1} \right)}}$
So, all the middle terms get canceled leaving the first and last term only.
Simplifying the expression, we get,
$ \Rightarrow \dfrac{{\left( {n + 1} \right) - 1}}{{\left( {n + 1} \right)}} = \dfrac{n}{{n + 1}}$

Option ‘B’ is correct

Note: We can split up a term into two terms if the denominator of both the terms are the same and the numerator consists of addition or subtraction of two terms. The rules of BODMAS must be applied while calculating and dealing with terms. The terms with the same magnitude and opposite signs get canceled.