Cadence, the pedal speed in cycling, is measured in pedal stroke revolutions per minute (RPM). For example, a cadence of 60 RPM means that one pedal makes a complete revolution 60 times in one minute. Every complete revolution of the pedal stroke of a stationary bicycle is equivalent to cycling an approximate distance of 0.006 km. Lisa signed up for a beginner spinning class. The workout routine requires her to start cycling at 30 RPM consistently for the first minute and progressively increase the cadence by 2.5 RPM for every subsequent minute. Celina signed up for an intermediate spinning class. The workout routine requires her to start cycling at 25 RPM consistently for the first minute and increase the cadence by 15% for every subsequent minute.

1. Set up an equation for the $n$-th term of the arithmetic progression.

Lisa's cadence forms an arithmetic progression with first term $a = 30$ and common difference $d = 2.5$.
Let $n$ be the minute at which she cycles at 120 RPM.
$u_n = a + (n-1)d$
$120 = 30 + (n-1)(2.5)$

2. Solve for $n$.

$120 - 30 = 2.5(n-1)$
$90 = 2.5(n-1)$
$n-1 = \frac{90}{2.5}$
$n-1 = 36$
$n = 37$

1. Find the total number of pedal revolutions in the first 15 minutes using the sum of an arithmetic progression.

$S_{15} = \frac{15}{2} \left[2(30) + (15 - 1)(2.5)\right]$
$S_{15} = 7.5 \left[60 + 35\right]$
$S_{15} = 712.5 \text{ revolutions}$

2. Calculate the total distance cycled in kilometers.

$\text{Total distance} = 712.5 \times 0.006$
$\text{Total distance} = 4.275 \text{ km}$

3. Convert the time to hours and calculate the average speed.

$\text{Time in hours} = \frac{15}{60} = 0.25 \text{ h}$
$\text{Average speed} = \frac{4.275}{0.25}$
$\text{Average speed} = 17.1 \text{ km/h}$

1. Formulate an inequality for the cadence at the $n$-th minute using the formula for the $n$-th term of a geometric sequence.

$25(1.15)^{n-1} > 120$

2. Solve the exponential inequality using logarithms to find $n$.

$1.15^{n-1} > 4.8$
$(n-1)\ln(1.15) > \ln(4.8)$
$n - 1 > \frac{\ln(4.8)}{\ln(1.15)}$
$n - 1 > 11.223$
$n > 12.223$

3. Identify the exact minute from the calculated value of $n$.

Since $n$ must be an integer representing the minute, $n = 13$.

1. Express the total distance cycled by Lisa in $n$ minutes

Lisa's cadence follows an arithmetic progression with first term $a = 30$ and common difference $d = 2.5$.
Total revolutions in $n$ minutes, $S_L = \frac{n}{2}[2(30) + (n-1)(2.5)]$
Total distance, $D_L = 0.006 \times \frac{n}{2}[60 + 2.5(n-1)]$

2. Express the total distance cycled by Celina in $n$ minutes

Celina's cadence follows a geometric progression with first term $a = 25$ and common ratio $r = 1.15$.
Total revolutions in $n$ minutes, $S_C = \frac{25(1.15^n - 1)}{1.15 - 1}$
Total distance, $D_C = 0.006 \times \frac{25(1.15^n - 1)}{0.15}$

3. Find the smallest integer $n$ for which Celina's distance exceeds Lisa's distance

We require $D_C > D_L$.
Comparing the values of $D_C$ and $D_L$ for integer values of $n$:
When $n = 5$, $D_C \approx 1.011 \text{ km} < D_L = 1.050 \text{ km}$
When $n = 6$, $D_C \approx 1.313 \text{ km} > D_L = 1.305 \text{ km}$
Since Celina's total distance cycled first exceeds Lisa's total distance at $n=6$, this occurs at the 6th minute.