Merging multiple summations into a single summation

Question

I'm doing a homework assignment which requires us to merge multiple summations into a single sum before computing it. The given problem is: $$\sum_{i=1}^{100} i^2 + \sum_{i=1}^{101} i + \sum_{i=k}^{102} k + \sum_{i=1}^{103} 1$$

I know the following properties of summation where:$$\sum_{k=m}^{n} a_k = \sum_{k=m}^{n-1} a_k + a_n$$ and $$\sum_{k=m}^{n} a_k + \sum_{k=m}^{n} b_k = \sum_{k=m}^{n} (a_k +b_k)$$

but I'm still having trouble using them to get a single summation. Help would be much appreciated.

Answer 1

I don't know if this is meant to be solved in this manner or not, but I don't think you can solve this in a conventional way. But you can also do this out of the way;

$$\sum_{i=1}^{103}\left(i^2\left(1-\left \lfloor{\frac{i}{101}}\right \rfloor\right) + i\left(1-\left \lfloor{\frac{i}{102}}\right \rfloor\right) +(k+i)\left(1-\left \lfloor{\frac{i}{102}}\right \rfloor\right)+ 1\right)$$

Good luck trying to compute from this.