Series expansion in a recurrence relation (Lines in a plane)

Question

L

The recurrence is therefore

L₀ = 1 ;

L_n = L_n−1 + n , for n > 0.

The known values of L₁ , L₂ , and L₃ check perfectly here, so we'll buy this. Now we need a closed-form solution. We could play the guessing game again, but 1 , 2 , 4 , 7 , 11 , 16 , . . . doesn't look familiar; so let's try another tack. We can often understand a recurrence by "unfolding" or "unwinding" it all the way to the end, as follows:

L_n = L_n-1 + n

= L_n−2 + (n − 1) + n

= L_n−3 + (n − 2) + (n − 1) + n

.

.

= L₀ + 1 + 2 + · · · + (n − 2) + (n − 1) + n

= 1 + S_n , where S n = 1 + 2 + 3 + · · · + (n − 1) + n.

In other words, L n is one more than the sum S n of the first n positive integers.

I didn't understand how

= L₀ + 1 + 2 + · · · + (n − 2) + (n − 1) + n is derived from previous step