Let value intially be = P
Then it is decreased by 20 %.
So 20% of P = ![\frac{20}{100} \times P = 0.2P](https://tex.z-dn.net/?f=%20%5Cfrac%7B20%7D%7B100%7D%20%5Ctimes%20P%20%3D%200.2P%20)
So after 1 year value is decreased by 0.2P
so value after 1 year will be = P - 0.2P (as its decreased so we will subtract 0.2P from original value P) = 0.8P-------------------------------------(1)
Similarly for 2nd year, this value 0.8P will again be decreased by 20 %
so 20% of 0.8P = ![\frac{20}{100} \times 0.8P = (0.2)(0.8P)](https://tex.z-dn.net/?f=%20%5Cfrac%7B20%7D%7B100%7D%20%5Ctimes%200.8P%20%3D%20%280.2%29%280.8P%29%20)
So after 2 years value is decreased by (0.2)(0.8P)
so value after 2 years will be = 0.8P - 0.2(0.8P)
taking 0.8P common out we get 0.8P(1-0.2)
= 0.8P(0.8)
-------------------------(2)
Similarly after 3 years, this value
will again be decreased by 20 %
so 20% of ![P(0.8)^{2} \frac{20}{100} \times P(0.8)^{2} = (0.2)P(0.8)^{2}](https://tex.z-dn.net/?f=%20P%280.8%29%5E%7B2%7D%20%20%5Cfrac%7B20%7D%7B100%7D%20%5Ctimes%20P%280.8%29%5E%7B2%7D%20%3D%20%280.2%29P%280.8%29%5E%7B2%7D%20)
So after 3 years value is decreased by ![(0.2)P(0.8)^{2}](https://tex.z-dn.net/?f=%20%280.2%29P%280.8%29%5E%7B2%7D%20)
so value after 3 years will be = ![P(0.8)^{2} - (0.2)P(0.8)^{2}](https://tex.z-dn.net/?f=%20P%280.8%29%5E%7B2%7D%20%20%20-%20%280.2%29P%280.8%29%5E%7B2%7D%20)
taking
common out we get ![P(0.8)^{2}(1-0.2)](https://tex.z-dn.net/?f=%20P%280.8%29%5E%7B2%7D%281-0.2%29%20)
![P(0.8)^{2}(0.8)](https://tex.z-dn.net/?f=%20P%280.8%29%5E%7B2%7D%280.8%29%20)
-----------------------(3)
so from (1), (2), (3) we can see the following pattern
value after 1 year is P(0.8) or ![P(0.8)^{1}](https://tex.z-dn.net/?f=%20P%280.8%29%5E%7B1%7D%20)
value after 2 years is ![P(0.8)^{2}](https://tex.z-dn.net/?f=%20P%280.8%29%5E%7B2%7D%20)
value after 3 years is ![P(0.8)^{3}](https://tex.z-dn.net/?f=%20P%280.8%29%5E%7B3%7D%20)
so value after x years will be
( whatever is the year, that is raised to power on 0.8)
So function is best described by exponential model
where y is the value after x years
so thats the final answer