The correct answer is the fourth option
The ordered pairs are:
<span>{(1, 80), (2, 240), (3, 720), (4, 2160)}
with
a_1 = 80
a_2 = 240
a_3 = 720
a_4 = 2160
The terms are in a geometric sequence with a common ratio of
240/80 = 720/240 = 2160/720 = 3
The equation is:
a_n = 80 * 3^(n - 1)</span>
It’s number 1 it’s a whole number because it has the same numerator and denominator
The present value (PV) of a loan for n years at r% compounded t times a year where there is equal P periodic payments is given by:
![PV=P\left( \frac{1-\left(1+ \frac{r}{t} \left)^{-nt}}{ \frac{r}{t} } \right)](https://tex.z-dn.net/?f=PV%3DP%5Cleft%28%20%5Cfrac%7B1-%5Cleft%281%2B%20%5Cfrac%7Br%7D%7Bt%7D%20%5Cleft%29%5E%7B-nt%7D%7D%7B%20%5Cfrac%7Br%7D%7Bt%7D%20%7D%20%5Cright%29)
Given that <span>Beth
is taking out a loan of PV = $50,000 to purchase a new home for n = 25 years at an interest rate of r = 14.25%. Since she is making the payment monthly, t = 12.
Her monthly payment is given by:
![50,000=P\left( \frac{1-\left(1+ \frac{0.1425}{12} \right)^{-25\times12}}{ \frac{0.1425}{12} } \right) \\ \\ =P\left( \frac{1-(1+0.011875)^{-300}}{ 0.011875 } \right)=P\left( \frac{1-(1.011875)^{-300}}{ 0.011875 } \right) \\ \\ =P\left( \frac{1-0.028969}{ 0.011875 } \right)=P\left( \frac{0.971031}{ 0.011875 } \right)=81.770994P \\ \\ \therefore P= \frac{50,000}{81.770994} =\$611.46](https://tex.z-dn.net/?f=50%2C000%3DP%5Cleft%28%20%5Cfrac%7B1-%5Cleft%281%2B%20%5Cfrac%7B0.1425%7D%7B12%7D%20%5Cright%29%5E%7B-25%5Ctimes12%7D%7D%7B%20%5Cfrac%7B0.1425%7D%7B12%7D%20%7D%20%5Cright%29%20%5C%5C%20%20%5C%5C%20%3DP%5Cleft%28%20%5Cfrac%7B1-%281%2B0.011875%29%5E%7B-300%7D%7D%7B%200.011875%20%7D%20%5Cright%29%3DP%5Cleft%28%20%5Cfrac%7B1-%281.011875%29%5E%7B-300%7D%7D%7B%200.011875%20%7D%20%5Cright%29%20%5C%5C%20%20%5C%5C%20%3DP%5Cleft%28%20%5Cfrac%7B1-0.028969%7D%7B%200.011875%20%7D%20%5Cright%29%3DP%5Cleft%28%20%5Cfrac%7B0.971031%7D%7B%200.011875%20%7D%20%5Cright%29%3D81.770994P%20%5C%5C%20%20%5C%5C%20%20%5Ctherefore%20P%3D%20%5Cfrac%7B50%2C000%7D%7B81.770994%7D%20%3D%5C%24611.46)
Therefore, her monthly payment is about $611.50
</span>