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Important Math Relations for Chemists

Important Math Relations for Chemists

by Admin User -
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Summation Formula: 

Constant Series

\begin{equation}
\sum_{k=0}^{S} c=s c
\end{equation}

Example: If S = 4

\begin{equation}
\begin{array}{c}{\sum_{k=0}^{4} c=0+c+c+c+c=4 c} \\ {\sum_{k=0}^{4} c=4 c}\end{array}
\end{equation}

Linear Series: \begin{equation}
\sum_{k=0}^{S} k=\frac{S(S+1)}{2}
\end{equation}

Example: If S = 4 \begin{equation}
\begin{array}{c}{\sum_{k=0}^{4} k=0+1+2+3+4=10} \\ {\sum_{k=0}^{S} k=\frac{S(S+1)}{2}=10}\end{array}
\end{equation}

Quadratic Series

\begin{equation}
\sum_{k=0}^{S} k^{2}=\frac{S(S+1)(2 S+1)}{6}
\end{equation}

Example if S = 4 \begin{equation}
\begin{array}{l}{0^{2}+1^{2}+2^{2}+3^{2}+4^{2}=0+1+4+9+16=30} \\ {\sum_{k=0}^{4} k^{2}=\frac{4(4+1)(2 \times 4+1)}{6}=\frac{4 \times 5 \times 9}{6}=30}\end{array}
\end{equation}

Roos of Quadratic Equation

If, \begin{equation}
a x^{2}+b x+c=0
\end{equation}

then, \begin{equation}
x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}
\end{equation}

Binomial Expansion

\begin{equation}
(1+x)^{n}=1+n x+\frac{n(n-1)}{2 !} x^{2}+\frac{n(n-1)(n-2)}{3 !} x^{3}+\ldots \ldots \ldots \ldots \ldots
\end{equation}

To compute \((a+b)^{n}\)

\begin{equation}
(a+b)^{n}=a^{n}(1+x)^{n}, \quad \text { with } x=\frac{b}{a}
\end{equation} or \begin{equation}
\circ(a+b)^{n}=b^{n}(1+y)^{n}, \quad \text { with } y=\frac{a}{b}
\end{equation}

\begin{equation}
(a+x)^{n}=\sum_{k=0}^{n}\left(\begin{array}{c}{n} \\ {k}\end{array}\right) x^{k} a^{n-k},\left(\begin{array}{c}{n} \\ {k}\end{array}\right)=\frac{n !}{k !(n-k) !}
\end{equation}

\begin{equation}
(1+r)^{-1}=1-r+r^{2}-r^{3}+\cdots
\end{equation}

\begin{equation}
a\left(\frac{1-r^{n}}{1-r}\right)=a+a r+a r^{2}+a r^{3}+\cdots
\end{equation}

\begin{equation}
\sum y^{i}=1+y+y^{2}+\cdots \cdots=\frac{1}{1-y},(y<1)
\end{equation}

\begin{equation}
\sum i y^{i}=y\left(1+2 y+3 y^{2}+\cdots \cdots\right)=\frac{y}{(1-y)^{2}},(y<1)
\end{equation}

Trigonometric Formula

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