Question

lucia and maria are business women who decided to invest money by buying farm land in brazil.

• lucia bought 11 hectares of land in the first month, and each month afterwards she buys 5 additional hectares.
• maria bought 6 hectares of land in the first month, and each month afterward her total number of hectares increases by a factor of 1.4.

they started their investments at the same time, and they both buy the additional land at the beginning of each month.
what is the first month in which marias amount of land exceeds lucias amount of land?

Explanation:

Step1: Define Lucia's land formula

Lucia's land follows an arithmetic sequence. Let $n$ = month number.
First term $a_1=11$, common difference $d=5$.
Formula: $L(n) = 11 + 5(n-1) = 5n + 6$

Step2: Define Maria's land formula

Maria's land follows a geometric sequence. First term $g_1=6$, common ratio $r=1.4$.
Formula: $M(n) = 6 \times (1.4)^{n-1}$

Step3: Test values of $n$

• $n=1$: $L(1)=11$, $M(1)=6$ → $M
• $n=2$: $L(2)=16$, $M(2)=8.4$ → $M
• $n=3$: $L(3)=21$, $M(3)=11.76$ → $M
• $n=4$: $L(4)=26$, $M(4)=16.464$ → $M
• $n=5$: $L(5)=31$, $M(5)=23.0496$ → $M
• $n=6$: $L(6)=36$, $M(6)=32.26944$ → $M
• $n=7$: $L(7)=41$, $M(7)=45.177216$ → $M>L$

Answer:

7