Given to two sums at the top, how am I supposed to calculate the following 5 problems? I've tried splitting sigma notation where possible but I still cant figure it out, any hints?
I have completed problems 1 and 2 but any assistance on problems 3, 4 and 5 would be appreciated!
Try these: 3. $$\sum_{k=2}^{21} a_{k-1} = \sum_{k=1}^{20} a_{k}$$ $$\sum_{k=2}^{21} a_{\lfloor k/2\rfloor} = \sum_{k=2,4,...,20} a_{\lfloor k/2\rfloor} + \sum_{k=3,5,...,21} a_{\lfloor k/2\rfloor} = \sum_{k=1}^{10} a_{k} + \sum_{k=1}^{10} a_{k}$$
4. $$\sum_{i=1}^{10}a_{2i} + \sum_{i=1}^{10}a_{2i-1} = \sum_{i=1}^{20}a_{i}$$
5. $$\sum_{i=1}^{10}(i+a_i) = \sum_{i=1}^{10} i + \sum_{i=1}^{10} a_i$$ and $$\sum_{i=1}^{10} i = \frac{(1+10)10}{2}$$