Alice and Bob started work on the same day. Alice's wage the first day was $5$ dollars, the next day (she is a good worker) it was $6$ dollars, the next day it was $7$ dollars, and so on. On the last day she worked, she earned $n$ dollars.
Note that this means her total income $A$ was given by
and she worked for $n-5+1$ days.
Bob's wage the first day was $n$ dollars. But the next day his wage was decreased by a dollar, and the same happened the day after that, and so on. On his last day he got $5$ dollars.
It is clear that Bob's total income was also $A$.
I forgot to mention that Alice and Bob are "partners." Every day, their joint income was $n+5$, since that was their joint income the first day, and every day Alice's income went up by $1$, and Bob's went down by $1$, leaving their combined daily income unchanged.
Between them, they earned $2A$ dollars. Every one of the $n-4$ days, they earned a combined $n+5$ dollars, so
It follows that
This gives us the desired closed form expression for the sum (1).
Remark: The same "story" can be used to find a closed form for the sum $a+(a+d)+(a+2d)+\cdots +(a+md)$.