# Arithmetic of series

#### steven187

hello all

a simple question how come

$$\sum_{k=1}^{n} k- \sum_{k=1}^{n} (k-1) \not= 1$$

even though i know if you expand it out you would get n, isnt there an arithmetic property of series that relates to this?

thanxs

#### whozum

The sum of the consecutive sequence of integers starting with 1 is

$$\frac{1}{2}n(n+1)$$ where n is the last (and largest) member of the sequence.

so from here we can see that the set of any consecutive set of integers has the sum given by

$$\frac{1}{2}n_1(n_1+1) - \frac{1}{2}n_2(n_2+1)$$, combining:

$$\frac{1}{2}(n_1^2 + n_1 - n_2^2 - n_2)$$

#### steven187

hello there

thanxs for that, i should have remembered something simple like that

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