determine the term of $2100 with an interest rate of 21.77% a compounded semi-annually if the future value is $5900

7 years
5 years
30 years
16 years

2100(1+.2177/2)^(2*t) = 5900

what is 5900 is years?

thank

I got 2582.05 idk what that is in years

clearly you have no idea what you are doing, david.

2100(1+.2177/2)^(2*t) = 5900
(1+.2177/2)^(2*t) = 2.8095
2t log(1.10885) = log(2.8095)
2t * 0.04487 = 0.44863
t = 4.999

so, it will take 5 years

To determine the term, or the number of years, needed to reach a future value of $5900 with an interest rate of 21.77% compounded semi-annually, we can use the formula for compound interest:

\[FV = PV \times \left(1+\frac{r}{n}\right)^{nt}\]

Where:
FV = Future value (given as $5900)
PV = Present value (initial amount invested)
r = Annual interest rate (given as 21.77% or 0.2177)
n = Number of times the interest is compounded per year (semi-annually means it is compounded twice a year, so n = 2)
t = Number of years

We can rearrange this formula to solve for t:

\[t = \frac{\log\left(\frac{FV}{PV}\right)}{n \times \log\left(1+\frac{r}{n}\right)}\]

Now let's plug in the values provided and calculate the terms:

For the first option (7 years):
\[t = \frac{\log\left(\frac{5900}{2100}\right)}{2 \times \log\left(1+\frac{0.2177}{2}\right)}\]
Calculating this equation gives us approximately 4.705 years, which is not equal to 7 years.

For the second option (5 years):
\[t = \frac{\log\left(\frac{5900}{2100}\right)}{2 \times \log\left(1+\frac{0.2177}{2}\right)}\]
Calculating this equation gives us approximately 3.514 years, which is not equal to 5 years.

For the third option (30 years):
\[t = \frac{\log\left(\frac{5900}{2100}\right)}{2 \times \log\left(1+\frac{0.2177}{2}\right)}\]
Calculating this equation gives us approximately 16.487 years, which is not equal to 30 years.

For the fourth option (16 years):
\[t = \frac{\log\left(\frac{5900}{2100}\right)}{2 \times \log\left(1+\frac{0.2177}{2}\right)}\]
Calculating this equation gives us approximately 9.876 years, which is not equal to 16 years.

None of the provided options match the calculated terms using the formula.