Basic mathematics refresher
An arithmetic progression is a sequence of numbers such that the difference from any term to its preceeding term is constant throughout the progression. Its individual terms are \(a\), \(a+d\), \(a+2d\), \(\cdots\), and the \(n^{th}\) term can be expressed as \(a_n=a_1 + (n-1)d\).
The sum \(S_n\) of an arithmetic progression with \(n\) terms is called an arithmetic series.
\[
\boxed{
\begin{gathered}
\\
\\
\quad S_n=\large{\frac{n(a_1+a_n)}{2}}\quad\\
\\
\end{gathered}
}
\]
A geometric progression is a sequence of numbers such that the relative difference from any term to its preceeding term is constant throughout the progression. Its individual terms are \(a\), \(ar\), \(ar^2\), \(\cdots\), and the \(n^{th}\) term can be expressed as \(a_n=a_1r^{n-1}\).
The sum \(S_n\) of a geometric progression with \(n\) terms is called a geometric series.
\[ \boxed{ \begin{gathered} \\ \\ \quad S_n=\large{a \cdot \frac{1-r^n}{1-r}}\quad\, {,} \quad r \neq 1 \quad \\ \\ \end{gathered} } \] If the ratio \(r\) is such taht \(-1 < r < 1\), then the sum of an infinite geometric progression as \(n\) approaches infinity is ginve by
\[
\boxed{
\begin{gathered}
\\
\\
\quad S_n=\large{\frac{a}{1-r}}\quad\, {,} \quad r \neq 1 \quad \\
\\
\end{gathered}
}
\]
\[
\boxed{
\begin{gathered}
\\
\\
\quad 1+2+ \cdots +n = {\frac{n(n+1)}{2}} \quad \\
\\
\quad 1^2+2^2+ \cdots +n^2 = {\frac{n(n+1)(2n+1)}{6}} \quad \\
\\
\quad 1^3+2^3+ \cdots +n^3 = {\frac{n^2(n+1)^2}{4}} \quad \\
\\
\end{gathered}
}
\]
\[ \boxed{ \begin{gathered} \\ \\ \quad f(x+h)=f(x)+hf'(x) + \frac{h^2}{2!}f{''}(x) + \frac{h^3}{3!}f{'''}(x) + \cdots \quad \\ \\ \end{gathered} } \]