This isn't as hard as it looks

Algebra Level 2

Find the value of $\frac{2}{5\times 7}+\frac{2}{7\times 9}+\frac{2}{9\times 11}+\cdots +\frac{2}{53\times 55}.$

\dfrac{54}{55}

\dfrac{55}{10}

\dfrac{53}{55}

\dfrac{2}{11}

1 solution

Kritarth Lohomi
Feb 1, 2015

Break every term

$\dfrac{2}{5\times7}= \dfrac{1}{5}-\dfrac{1}{7}$

Similarly break all terms alternate terms cancel out

to get

$\dfrac{1}{5}-\dfrac{1}{55}= \boxed{\dfrac{2}{11}}$

Upvote if you are satisfied

A more formal way would be to express the given sum in summation notation as follows:

$\sum_{n=3}^{27} \left( \dfrac{2}{(2n-1)(2n+1)}\right)$

This reduces to,

$\sum_{n=3}^{27} \left( \dfrac{1}{2n-1}-\dfrac{1}{2n+1}\right)$

This is a telescoping sum where all the middle terms would get cancelled out and the sum would reduce to: $\dfrac{1}{5}-\dfrac{1}{55}$

The rest is the same as yours.

Prasun Biswas - 6 years, 4 months ago