Find $\sum^{30}_{k=7}(\sqrt{k - 4} - \sqrt{k - 3})$ without explicitly calculating the 24 complements

Question

Is it possible to find $\sum^{30}_{k=7}(\sqrt{k - 4} - \sqrt{k - 3})$ without explicitly calculating the 24 complements ?

I have this task in a text book and it doesn't provide any equivalent example, which makes me think that the only way to solve it is by explicitly calculating the complements ?

Thank you in advance

Answer 1

HINT

We have that

$$\sum^{30}_{k=7}(\sqrt{k - 4} - \sqrt{k - 3})=\sum^{30}_{k=7}\sqrt{k - 4} - \sum^{30}_{k=7}\sqrt{k - 3}=\sum^{30}_{k=7}\sqrt{k - 4} - \sum^{31}_{j=8}\sqrt{j - 4}$$