Arithmetic Sequences Problems

Question

Find the sum $S$ defined by $$S = \sum_{n=1}^{20} \left(3n-\frac{ 1}{2}\right).$$

I have

$$S = 3 \sum_{n=1}^{20} n- \sum_{n=1}^{20}\frac{ 1}{2} = 3(210) - 10 = 620,$$

but the answer is supposed to be 1380?

Answer 1

The right answer is $S=620$.

$$\begin{align} S&=\sum_{n=1}^{20}\Big(3n-\frac12\Big) \\ &=3\sum_{n=1}^{20}n-\sum_{n=1}^{20}\frac12 \\ &=3\frac{20\cdot21}{2}-20\cdot\frac12 \\ &=3\cdot210-10 \\ &=\boxed{620} \end{align}$$