$$\sum_{i=0}^{364}{1+i}$$
You asked:
Evaluate the expression: \(\sum_{i = 0}^{364} {1 + i}\)
MathBot Answer:
\[\sum_{i=0}^{364} \left(i + 1\right) = 66795\]
$$\begin{aligned}\sum_{i=0}^{364} \left(i + 1\right) &= \sum_{i=0}^{364} 1 + \sum_{i=0}^{364} i \ \ \ \small{\color{grey}\text{by the identity} \sum_{n=k}^{m} \left(f{\left(n \right)} + g{\left(n \right)}\right) = \sum_{n=k}^{m} f{\left(n \right)} + \sum_{n=k}^{m} g{\left(n \right)}}\\&= \sum_{i=0}^{364} i + 1 \cdot 365 \ \ \ \small{\color{grey}\text{by the identity} \sum_{n=0}^{m} C = C(m+1)}\\&= \sum_{i=0}^{364} i + 365\\&= \frac{364 \cdot \left(1 + 364\right)}{2} + 365 \ \ \ \small{\color{grey}\text{by the identity} \sum_{n=0}^{m} n = \frac{m(m+1)}{2}}\\&= 365 + 66430\\&= 66795\end{aligned}$$