Tristian Velazquez

2022-06-22

Confusing sum of fractions

Question is to find the sum of:

$(\frac{1}{{2}^{2}-1})+(\frac{1}{{4}^{2}-1})+(\frac{1}{{6}^{2}-1})+(\frac{1}{{20}^{2}-1})$

I know that ${a}^{2}-{b}^{2}=(a+b)(a-b)$ , and that with this I can find the LCM to be 1995, but this isn't helping me find the answer. I am looking for some kind of formula, because this question can come in an exam (having 60 qs for 60 mins), so they must not be giving a brute-force approach question.

Please help me in finding a fast approach.

Thanks

Question is to find the sum of:

$(\frac{1}{{2}^{2}-1})+(\frac{1}{{4}^{2}-1})+(\frac{1}{{6}^{2}-1})+(\frac{1}{{20}^{2}-1})$

I know that ${a}^{2}-{b}^{2}=(a+b)(a-b)$ , and that with this I can find the LCM to be 1995, but this isn't helping me find the answer. I am looking for some kind of formula, because this question can come in an exam (having 60 qs for 60 mins), so they must not be giving a brute-force approach question.

Please help me in finding a fast approach.

Thanks

Carmelo Payne

Beginner2022-06-23Added 25 answers

It's also true that

$\frac{1}{(a+b)(a-b)}=\frac{1}{2b}(\frac{1}{a-b}-\frac{1}{a+b})$

Thus you can write your sum as (choosing $b=1$ in every term)

$\frac{1}{2}(\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{19}-\frac{1}{21})$

$\frac{1}{(a+b)(a-b)}=\frac{1}{2b}(\frac{1}{a-b}-\frac{1}{a+b})$

Thus you can write your sum as (choosing $b=1$ in every term)

$\frac{1}{2}(\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{19}-\frac{1}{21})$