Fun with series 3

sum of first natural numbers is given by n(n+1)/2 there the series is $1 - 2( \frac{1}{2.3} + \frac{1}{3.4} + .............. + \frac{1}{50.51})$ now we can write

$\frac{1}{2.3}$ as $\frac{1}{2} -\frac{1}{3}$

similarly every term and at last we get $\frac{ 2}{51}$ m +n =53

The denominator of every successive odd term is increases by 1 in the series. Let T(x) denote the xth term and S(x) the sum of x terms. By calculation of the sum of the first few terms, we know that S(1)=1, S(3)=1/2, S(5)=1/3. Let d denote the denominator of any odd term in the series. Notice that the relation between S(x) and d is: S(x)=2d-1. Thus solving for S(49) would give the value d=25. Therefore S(49) is: 1/25. To find the denominator of the last term T(50), we use the formula for the sum of an arithmetic progression: (50/2)(2+(50-1))=1275. Thus S(50) =(1/25)-(1/1275)=2/51. And 2+51=53!