Sum of sums

$\displaystyle \sum_{i = 1}^{100} \sum_{j = 1}^{i} ( 1 + 1 + ..... j times)$

$\displaystyle \sum_{i = 1}^{100}( 1 + 2 + 3 +...... + i) = \frac{(i)(i + 1)}{2} = \frac{i^{2} + i}{2}$

$\frac{1}{2}(\displaystyle \sum_{i = 1}^{100} i^{2} + \displaystyle \sum_ {i=1}^{100}i)$

$\frac{1}{2}(338350 + 5050) = 171700$

$\sum _{i=1} ^{100} {\sum _{j=1} ^{i} {\sum _{k=1} ^{j} {(x^{0\times k})}}} = \sum _{i=1} ^{100} {\sum _{j=1} ^{i} {j}} = \sum _{i=1} ^{100} {\frac {i (i+1)}{2}}$ $= \frac {1}{2} \left( \frac {100\times 101\times 201} {6} + \frac {100\times 101}{2} \right)$ $= \frac {1}{2} \times \left( 338350 + 5050 \right) = \boxed {171700}$