$\frac{6}{4 \times 2} + \frac{7}{5 \times 2} + ... + \frac{21}{19 \times 2}$

Question

I got this exercise from school and I have no idea what notion to use, it resumes to Harmonic series, I can't find a generic answer. Do you have any idea?

$\frac{6}{4 \times 2} + \frac{7}{5 \times 2} + ... + \frac{21}{19 \times 2}$

Which is the generic answer for such a sum?

Answer 1

Consider the most general case $$A=\sum_{i=a}^b\frac n{2(n-2)}=\frac 12\sum_{i=a}^b\frac n{n-2}=\frac 12\sum_{i=a}^b\frac {n-2+2}{n-2}$$ $$A=\frac 12\sum_{i=a}^b1+\sum_{i=a}^b\frac {1}{n-2}=(b-a+1)+\sum_{i=a}^b\frac {1}{n-2}$$ So, you are just left with the sum of fractions $\frac 14+\frac 15+\frac 16+\cdots+\frac 1{19}$

Answer 2

We got the following sum:

$$\sum\limits _{n=6}^{21}\frac{n}{2(n-2)}=$$

$$\frac{6}{2(6-2)}+\frac{7}{2(7-2)}+\frac{8}{2(8-2)}+\frac{9}{2(9-2)}+...+\frac{21}{2(21-2)}=$$

$$\frac{6}{8}+\frac{7}{10}+\frac{8}{12}+\frac{9}{14}+...+\frac{21}{38}=$$

$$\frac{753813839}{77597520}$$