Reference Materials for Mathematics

\begin{align*} \text{Shifting: } \sum_{k=0}^n a_k \amp = \sum_{k=r}^{n+r} a_{k-r} = \sum_{k=-r}^{n-r} a_{k+r} \\ \text{Removing Initial Terms: }\sum_{k=0}^n a_k \amp = a_0 + a_1 + \ldots + a_r + \sum_{k=r+1}^n a_k \end{align*}

\begin{align*} \sum_{k=1}^n 1 \amp = n \\[1em]\\ \sum_{k=1}^n k \amp = \frac{n(n+1)}{2} \\[1em]\\ \sum_{k=1}^n k^2 \amp = \frac{n(n+1)(2n+1)}{6} \\[1em]\\ \sum_{k=1}^n k^3 \amp = \left(\frac{n(n+1)}{2} \right)^2 \end{align*}