What is the sum $\sum_{n=n_0-N+l}^{n_0+N}1$?

Question

To be very concise, I am doing mechanics and, to my surprise,

$$\sum_{n=n_0-N+l}^{n_0+N}1 = 2N+1-l$$

Why is it so? Any help clarifying this would be very appreciated.

Answer 1

As the input of the summation is simply $1$ this effectively means that $$\sum_{n=a}^b1=\overbrace{1+1+1+...+1+1+1}^{b-a+1 \text{ times}}=b-a+1$$ So in your example $$\sum_{n=n_0-N+l}^{n_0+N}1=n_0+N-(n_0-N+l)+1=2N-l+1$$