Question

Find the sum S defined by
S = \sum_{n=1}^{20}(3n - 1 / 2)

Answer 1

$\begin{aligned}S\ \ &=\sum_{n=1}^{20}\frac{3n-1}{2}\\ \\ \Longrightarrow\ \ &\frac{1}{2}\sum_{n=1}^{20}3n-1\\ \\ \Longrightarrow\ \ &\frac{1}{2}\sum_{n=1}^{20}3n-\frac{1}{2}\sum_{n=1}^{20}1\\ \\ \Longrightarrow\ \ &\frac{3}{2}\sum_{n=1}^{20}n-\frac{1}{2}\sum_{n=1}^{20}1\end{aligned}$

$\begin{aligned}\Longrightarrow\ \ &\frac{3}{2}\bigg(1+2+3+...+20\bigg)-\frac{1}{2}\bigg(\underbrace{1+1+1+...+1}_{20}\bigg)\\ \\ \Longrightarrow\ \ &\frac{3}{2}\cdot\frac{20\cdot 21}{2}-\frac{1}{2}\cdot 20\\ \\ \Longrightarrow\ \ &315-10\\ \\ \Longrightarrow\ \ &\large\bold{305}\end{aligned}$

$\textsf{Hence,}\\ \\ \\ \Large\boxed{\textbf{S=305}}$