Trust takes time

Trust is built like a wall, one brick a day. Each day you keep your word, one brick is added.

Trust grows by one brick for each day I keep my word.

Trust $T(d)=d$, with $d$ honest days.

After forty days of kept promises you have forty bricks. Building them took forty days.

After $40$ days: $T=40$ bricks.

$T(40)=40$.

Now tell one lie. A lie does not take away one brick: it knocks down half the wall. Forty bricks become twenty in a single moment, and to get back to forty you need twenty more honest days.

One lie halves it: $T=40÷2=20$. Repairing it takes $40-20=20$ days.

Lie: $T=40/2=20$; repair $40-20=20$ days.

Someone who lies every ten days never builds a tall wall. After the first lie five bricks are left, after the second seven and a half, after the third eight and three quarters. The number creeps closer to ten, but never gets there, and before any lie the wall is never taller than twenty.

Suppose someone lies every $10$ days, and $x_n$ is the number of bricks after the $n$th lie. Then $x_0=0$ and $x_{n+1}=\frac{x_n+10}{2}$, so $x_1=5,x_2=7.5,x_3=8.75$. Always $x_n<10$, though $x_n\to 10$; just before a lie there are $x_n+10<20$ bricks.

$x_0=0$ and $x_{n+1}=(x_n+10)/2$, so $x_n=10(1-2^{-n})<10$, and ${\lim }_{n\to \infty }{x}_n=10$.

Building took forty days, losing half took a moment, repairing takes twenty days. Keep your word: it is the cheapest brick.

To build: $40$ days. To lose $20$ bricks: one moment. To repair: $20$ days.

Build $b=40$, lose $l=1$ moment, repair $r=20$ days.