Answer:
A. 1,500
Step-by-step explanation:
[tex]\boxed{\begin{minipage}{7cm}\underline{Sum of the first n terms of a geometric series}\\\\$S_n=\dfrac{a(1-r^n)}{1-r}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\ \phantom{ww}$\bullet$ $r$ is the common ratio.\\ \phantom{ww}$\bullet$ $n$ is the $n$th term.\\\end{minipage}}[/tex]
The given scenario can be modelled as a geometric series.
If Marcus' goal is to sell 15,000 t-shirts during the first 6 months, then:
If he projects that the number of t-shirts he sells will increase by 20% each month then the common ratio is:
Substitute these values into the formula and solve for a:
[tex]\implies 15000=\dfrac{a(1-1.2^6)}{1-1.2}[/tex]
[tex]\implies 15000(1-1.2)=a(1-1.2^6)[/tex]
[tex]\implies a=\dfrac{15000(1-1.2)}{1-1.2^6}[/tex]
[tex]\implies a=1510.586188[/tex]
Therefore, the approximate number of T-shirts Marcus needs to sell during the first month to meet his goal is 1,500.
Check:
- Month 1 = 1511
- Month 2 = 1511 × 1.2 = 1813
- Month 3 = 1813 × 1.2 = 2176
- Month 4 = 2176 × 1.2 = 2611
- Month 5 = 2611 × 1.2 = 3133
- Month 6 = 3133 × 1.2 = 3760
Total = 1511 + 1813 + 2176 + 2611 + 3133 + 3760 = 15004
Check:
- Month 1 = 1500
- Month 2 = 1500 × 1.2 = 1800
- Month 3 = 1800 × 1.2 = 2160
- Month 4 = 2160 × 1.2 = 2592
- Month 5 = 2592 × 1.2 = 3110
- Month 6 = 3110 × 1.2 = 3732
Total = 1500 + 1800 + 2160 + 2592 + 3110 + 3732 = 14894 ≈ 15000
