Question

the towns of franklin and chester post their populations on signs just outside of their towns. the signs are updated once a year at the beginning of the year.
during year 1, franklins sign read \franklin: population of 20,000\, while chesters sign read \chester: population of 25,000\. each year, the populations grew. specifically, franklins population grew by 5% each year, and chesters population grew by 500 people each year.
what is the first year in which franklins sign shows a larger number than chesters sign?
year \boxed{}

Explanation:

Step1: Define population functions

Let $n$ = number of years after Year 1.
Franklin's population: $F(n) = 20000 \times (1.05)^n$
Chester's population: $C(n) = 25000 + 500n$
We need $F(n) > C(n)$.

Step2: Test year by year (n=0)

Year 1 ($n=0$):
$F(0) = 20000 \times 1 = 20000$
$C(0) = 25000 + 0 = 25000$
$20000 < 25000$ (not larger)

Step3: Test n=1 (Year 2)

$F(1) = 20000 \times 1.05 = 21000$
$C(1) = 25000 + 500 = 25500$
$21000 < 25500$ (not larger)

Step4: Test n=2 (Year 3)

$F(2) = 21000 \times 1.05 = 22050$
$C(2) = 25500 + 500 = 26000$
$22050 < 26000$ (not larger)

Step5: Test n=3 (Year 4)

$F(3) = 22050 \times 1.05 = 23152.5$
$C(3) = 26000 + 500 = 26500$
$23152.5 < 26500$ (not larger)

Step6: Test n=4 (Year 5)

$F(4) = 23152.5 \times 1.05 = 24310.125$
$C(4) = 26500 + 500 = 27000$
$24310.125 < 27000$ (not larger)

Step7: Test n=5 (Year 6)

$F(5) = 24310.125 \times 1.05 = 25525.63125$
$C(5) = 27000 + 500 = 27500$
$25525.63 < 27500$ (not larger)

Step8: Test n=6 (Year 7)

$F(6) = 25525.63125 \times 1.05 = 26801.91281$
$C(6) = 27500 + 500 = 28000$
$26801.91 < 28000$ (not larger)

Step9: Test n=7 (Year 8)

$F(7) = 26801.91281 \times 1.05 = 28142.00845$
$C(7) = 28000 + 500 = 28500$
$28142.01 < 28500$ (not larger)

Step10: Test n=8 (Year 9)

$F(8) = 28142.00845 \times 1.05 = 29549.10887$
$C(8) = 28500 + 500 = 29000$
$29549.11 > 29000$ (larger)

Answer:

9