Can I derive a closed form for the n+1 sequence defined by x_n+1 = x_n+n?

[aaaa202]

For a system I am studying the following sequence (which I would assume is quite common) came up:

n1=1, n2=2, n3=4, n4=7, n5=11, n6=16, n7=22 ... i.e. the difference betweens two successive numbers grows with 1 as we move from (n_N-1, n_N) to (n_N,n_N+1).
Is there a closed form expression f(k) for this sequence, i.e. f(1)=n1, f(2)=n2, f(3)=n3 etc.

edit: So basically I have a sequence with I think what is called a recurence relation equal to:

x_n+1 = x_n+n

Can I find a closed form for this?

 

[pasmith]

For a system I am studying the following sequence (which I would assume is quite common) came up:

n1=1, n2=2, n3=4, n4=7, n5=11, n6=16, n7=22 ... i.e. the difference betweens two successive numbers grows with 1 as we move from (n_N-1, n_N) to (n_N,n_N+1).
Is there a closed form expression f(k) for this sequence, i.e. f(1)=n1, f(2)=n2, f(3)=n3 etc.

edit: So basically I have a sequence with I think what is called a recurence relation equal to:

x_n+1 = x_n+n

Can I find a closed form for this?

Hint:
$$\sum_{k=1}^n k - \sum_{k=1}^{n-1} k = n$$

The sum can be expressed in closed form as a standard result.