Zeno's Paradoxical 5K

I’ve been experimenting with different strategies in $5000$-meter races (known as “5K”s). If I run the distance at a steady pace, I’ll finish in precisely $23$ minutes. However, I tend to find a burst of energy as I near the finish line. Therefore, I’ve tried intentionally running what’s called a “negative split,” meaning I run the second half of the race faster than the first half—as opposed to a “positive split,” where I run a slower second half.

I want to take the concept of a negative split to the next level. My plan for an upcoming race—the “Zeno Paradox 5K”—is to start out with a $24$-minute pace (i.e., running at a speed such that if I ran the whole distance at that speed, I’d finish in $24$ minutes). Halfway through the race by distance (i.e., after $2500$ meters), I’ll increase my speed by $10$ percent. Three-quarters of the way through, I’ll increase by another $10$ percent. If you’re keeping track, that’s now $21$ percent faster than my speed at the start.

I continue in this fashion, upping my tempo by $10%$ every time I’m half the distance to the finish line from my previous change in pace. (Let’s put aside the fact that my speed will have surpassed the speed of light somewhere near the finish line.) Using this strategy, how long will it take me to complete the 5K? I’m really hoping it’s faster than my steady $23$-minute pace, even though I start out slower (at a $24$-minute pace).

Let the race start at $x=0$ and finish at $x=5.$ Let $x_n = 5(1 - 2^{-n})$ for $n = 0, 1, 2, \dots$ be the breakpoints where you are increasing your speed. Let $v(x)$ denote the velocity you are running (in kmph), when you are at some $x \in (0,5).$ Then $$v(x) = 12.5 \cdot 1.1^{\max \{ i : x_i \lt x \} }.$$ The time spent at the velocity $v_n = 12.5 \cdot 1.1^n$ is $$t_n = \frac{ x_{n+1}-x_n }{ v_n } = \frac{ 5 \cdot 2^{-(n+1)} }{ 12.5 \cdot 1.1^n } = \frac{2.2^{-n}}{5},$$ so the total time to complete the 5K in this fashion is $$T = \sum_{n=0}^\infty \frac{2.2^{-n}}{5} = \frac{1}{5} \frac{1}{1- \frac{1}{2.2}} = \frac{2.2}{5 \cdot 1.2} = \frac{11}{30}\,\,\text{ hours},$$ or $22$ minutes.