Çözüldü Geometric Sequence - Binomial Expansion

https://i72.servimg.com/u/f72/19/97/10/39/deprec10.png
https://file.recipes999.com/077/9781260454208.pdf
(Page 379, Problem 43.14)

Algebraic Solution:
x: Sale Price
r: Depreciation Rate

Value at the end of the first year:
x - x·r = (-1)^1·x·(r - 1)^1

Value at the end of the second year:
(x - x·r) - (x - x·r)·r = x·r^2 - 2x·r + x = (-1)^2·x·(r - 1)^2

Value at the end of the third year:
(x·r^2 - 2x·r + x) - (x·r^2 - 2x·r + x)·r = -x·r^3 + 2x·r^2 - x·r + x·r^2 - 2x·r + x = -x·r^3 + 3x·r^2 - 3x·r + x =
(-1)^3·x·(r - 1)^3

Value at the end of the fourth year:
(-x·r^3 + 3x·r^2 - 3x·r + x) - (-x·r^3 + 3x·r^2 - 3x·r + x)·r =
-x·r^3 + 3x·r^2 - 3x·r + x + x·r^4 - 3x·r^3 + 3x·r^2 - x·r = x·r^4 - 4x·r^3 + 6x·r^2 - 4x·r + x =
(-1)^4·x·(r - 1)^4

Value at the end of the fifth year:
(x·r^4 - 4x·r^3 + 6x·r^2 - 4x·r + x ) - (x·r^4 - 4x·r^3 + 6x·r^2 - 4x·r + x)·r =
-x·r^5 + 5x·r^4 - 10x·r^3 + 10x·r^2 - 5x·r + x = (-1)^5·x·(r - 1)^5

x = $87,500
r = 0.3
(-1)^5·$87,500·(0.3 - 1)^5 =
$87,500·(0.7^5) ≈
$14,706.125