Respuesta :
a) We know that each year the price is increased by 14%. That is that the next year price is 1.14 times the actual price.
We can write this as:
[tex]P_{n+1}=1.14\cdot P_n[/tex]This is a recursive formula because the price depends on the previous prices.
b) If we want to find a explicit formula in function of the years, we need to start from a known price value.
Then, by the recursive formula from the previous point, we can write:
[tex]\begin{gathered} P_1=1.14\cdot P_0 \\ P_2=1.14\cdot P_1=1.14\cdot(1.14\cdot P_0)=1.14^2\cdot P_0 \\ P_3=1.14\cdot P_2=1.14\cdot(1.14^2\cdot P_0)=1.14^3\cdot P_0 \end{gathered}[/tex]Following that reasoning we can write the explicit formula as:
[tex]P_n=1.14^n\cdot P_0[/tex]c) If the price for 2020 is P0=350, we can use the explicit formula to calculate the value for 2025, that is P5:
[tex]P_5=1.14^5\cdot P_0=1.14^5\cdot350\approx1.925\cdot350\approx673.90_{}[/tex]For year 2025, the predicted price is $673.90.
d) We can find when the price will reach $1000 by finding the value of n from the explicit formula.
We can write:
[tex]\begin{gathered} P_n=1.14^n\cdot P_0=1.14^n\cdot350=1000 \\ 1.14^n=\frac{1000}{350} \\ n\cdot\ln (1.14)=\ln (1000)-\ln (350) \\ n=\frac{\ln (1000)-\ln (350)}{\ln (1.14)} \\ n\approx\frac{6.908-5.858}{0.131}\approx\frac{1.050}{0.131}\approx8.012 \end{gathered}[/tex]The value for n is 8.012.
As n=0 correspond to 2020, the year where it will reach a price of $1000 is Year=2020+8.012=2028.012.
Answer: Year = 2028.012.
