$\sum_{k=1}^{100}{1} + \sum_{k=1}^{100}{k}$
You asked:
Evaluate the expression: \(\sum_{k = 1}^{100} {1 + \sum_{k = 1}^{100} {k}}\)
MathBot Answer:
\[\sum_{k=1}^{100} \left(1 + \sum_{k=1}^{100} k\right) = 505100\]
$$\begin{aligned}\sum_{k=1}^{100} \left(1 + \sum_{k=1}^{100} k\right) &= 100 \cdot \left(1 + \sum_{k=1}^{100} k\right) \ \ \ \small{\color{grey}\text{by the identity} \sum_{n=1}^{m} C = mC}\\&= 100 \cdot \left(1 + \frac{100 \cdot \left(1 + 100\right)}{2}\right) \ \ \ \small{\color{grey}\text{by the identity} \sum_{n=1}^{m} n = \frac{m(m+1)}{2}}\\&= 100 \cdot 5051\\&= 505100\end{aligned}$$