Caleb is training for a vertical marathon, by running up the stairs of a building with 62 floors. He decides to train three days a week starting with 10 floors on his first day of training. On each subsequent training day, he runs 2 floors more than the previous training day. Once he can complete 62 floors in a training day, he is considered as ready for the vertical marathon. Then for subsequent training days, he will continue to complete 62 floors each day. After training, Caleb's heart rate decreases by 10% at the start of every $n$th minute. When $n = 1$, his heart rate at the start of the first minute is 180 beats per minute (bpm). Caleb's friend, Alex also tracks his own heart rate after training. His heart rate at the start of the first minute is 170 bpm and his heart rate can be modelled by the recurrence relation $H_{n+1} = 0.85 H_n + 10.5$, where $H_n$ is his heart rate at the start of the $n$th minute after training.

1. Identify the parameters of the arithmetic progression

Let $u_n$ be the number of floors completed on the $n$-th training day.
The sequence of floors forms an arithmetic progression where the first term $a = 10$ and the common difference $d = 2$.

2. Calculate the number of floors on the 10th training day

$u_{10} = a + (10 - 1)d$
$u_{10} = 10 + 9(2)$
$u_{10} = 28$

1. Determine the training day on which Caleb completes 62 floors using the formula for the $n$th term of an arithmetic progression.

$u_n = a + (n-1)d$
$62 = 10 + (n-1)2$
$52 = 2(n-1)$
$n - 1 = 26 \implies n = 27$

2. Calculate the number of weeks $k$ needed to complete 27 training days.

Since he trains 3 days a week:
$k = \frac{27}{3} = 9$

1. Find the training day on which Caleb first completes 62 floors.

Let $n$ be the number of training days to reach 62 floors. Using the formula for the $n$th term of an arithmetic progression $u_n = a + (n - 1)d$:
$10 + (n - 1)(2) = 62$
$2(n - 1) = 52$
$n - 1 = 26$
$n = 27$

2. Calculate the total number of floors completed in the first 27 training days.

Using the sum formula for an arithmetic series $S_n = \frac{n}{2}(a + l)$:
$S_{27} = \frac{27}{2}(10 + 62)$
$S_{27} = \frac{27}{2}(72) = 972$

3. Calculate the overall total number of floors completed at the end of 20 weeks.

Total training days in 20 weeks = $20 \times 3 = 60$ days.
Remaining training days where he completes 62 floors = $60 - 27 = 33$ days.
Overall total floors completed = $972 + (33 \times 62)$
$= 972 + 2046 = 3018$

1. Formulate the expression for Caleb's heart rate at the start of the $n$th minute.

Let $C_n$ be Caleb's heart rate at the start of the $n$th minute.
Since it decreases by $10\%$ each minute, $C_n$ forms a geometric sequence with first term $a = 180$ and common ratio $r = 1 - 0.10 = 0.9$.
$C_n = 180(0.9)^{n-1}$

2. Set up the inequality for the heart rate to drop below 82 bpm.

$180(0.9)^{n-1} < 82$
$0.9^{n-1} < \frac{82}{180}$
$0.9^{n-1} < \frac{41}{90}$

3. Solve the inequality for $n$ using logarithms to find the least integer value.

Taking the natural logarithm on both sides:
$(n-1)\ln(0.9) < \ln\left(\frac{41}{90}\right)$
Since $\ln(0.9) < 0$, the inequality sign flips when dividing:
$n-1 > \frac{\ln(41/90)}{\ln(0.9)}$
$n-1 > 7.462$
$n > 8.462$
The least integer value of $n$ is $9$.

1. Generate the sequence using a graphing calculator to find terms around the threshold of 90 bpm.

Using a graphing calculator to generate the sequence $H_{n+1} = 0.85H_n + 10.5$ with $H_1 = 170$:
When $n = 10$, $H_{10} \approx 93.16 > 90$
When $n = 11$, $H_{11} \approx 89.69 < 90$

2. Identify the least value of $n$.

Since $H_{11}$ is the first term that falls below $90$ bpm, the least value of $n$ is $11$.

1. Set up the equation for the steady-state (resting) heart rate as $n \to \infty$.

After resting long enough, $n \to \infty$ and the heart rate will reach a steady value $L$.
As $H_{n+1} \to L$ and $H_n \to L$, substitute $L$ into Alex's recurrence relation:
$L = 0.85L + 10.5$

2. Solve the linear equation to find the resting heart rate.

$L - 0.85L = 10.5$
$0.15L = 10.5$
$L = \frac{10.5}{0.15} = 70$