sum of the first terms of an arithmetic progression

If we add the k first terms of an arithmetic progression, we can check that the addition of the first term plus the last is the same as the outcome you get when you add the second and the penultimate terms. Then we can easily obtain the next formula:

For example: the addition of the first ten terms of the progression: 1, 3, 5, 7, .. a_n= 2n – 1, is:

Exercises:

1.- Calculate the sum of the ten first terms of these arithmetic progressions:

a) 1, -1, -3, -5,...

b) a_n = 3n - 1

c) a₁ = 15, a₇ = 27

2.- If the sum 1 + 2 + 3 + 4 +... + n equals 60378. How many numbers did we add?

3.- Calculate the distance that a gardener covers when he pours a bucket of water from a well onto each of the 30 trees lined up in a row, knowing that the distance from the well to the first tree is 10 m and the distance between one tree and the next is 6 m, and considering he have to put the bucket back next to the well when he finishes.

Solutions: 1) a) -80; b) 165; c) 240; 2) n = 347: 3) 5820 m