The question describes a scenario of an increase of 3% every 6 months copounded monthly with a first term of 100.
To solve this question, we use the future value at compound interest formula noting that here 1 period is 6 months with monthly compounding (i.e. 6 compounding per period).
Recall that the future value at compound interest is given by
[tex]FV=PV(1+ \frac{r}{t} )^{nt}[/tex]
where: PV is the sale made in the first month = $100, r is the increase per period = 3%, t is the number of compounding every month = 6, n is the number of periods = 8 x 6 = 48 (8 periods of 6 months each).
[tex]FV=100\left(1+ \frac{0.03}{6} \right)^{(48+1)}[/tex]
Here, we added 1 to the number of period because we are interested in the first month after the fourth year.
[tex]FV=100\left(1+ 0.005 \right)^{49} \\ \\ =100(1.005)^{49} \\ \\ =100(1.276841607)=128[/tex] to the nearest whole number.
Therefore, 128 kits was sold in the first month of this year.
