Algebra: Number of layers in the pile of posts

The topmost layer in the pile contains 1 pole, the 2nd layer 2 poles, the third layers 3 poles, and so on. The bottom-most layer, say n^th layer, contains n posts.
$1 + 2 + 3 ~ + ... + ~ 5 ~ + ... + ~ n$
 

The number of poles in these layers of piles is in the form of arithmetic progression (AP) with a common difference d of 1. The first term a₁ in AP is 1 (for topmost layer) and the number of terms in AP is the number of layers in the pile. The formula is given by
$S = \frac{1}{2}n[ \, 2a_1 + (n - 1)d \, ]$

$1275 = \frac{1}{2}n[ \, 2(1) + (n - 1)(1) \, ]$

$2550 = 2n + n(n - 1)$

$2550 = 2n + n^2 - n$

$n^2 - n - 2550 = 0$

$(n - 51)(n + 50) = 0$

$n = 51 ~ \text{and} ~ -50$
 

There are 51 layers in the pile.