To one who practises

Dear one, you told me practice feels like nothing: twenty minutes a day and it seems that nothing changes. I am writing to tell you what that nothing hides.

Twenty minutes a day, and it seems nothing changes.

$20$ min a day.

Suppose each week of practice makes you five in a hundred better than the week before. After ten weeks you are not fifty in a hundred better but about sixty-three, because the gains are built on top of each other.

Each week your skill is multiplied by $1.05$. After $10$ weeks, $1.05^{10}\approx 1.63$, about $63$ per cent better, not $50$.

Skill $s_n=1.05^n$, $s_{10}\approx 1.63$.

After fourteen weeks you are still under double, at about 1.98 times where you began. After fifteen weeks you are at 2.08 times, and you have passed double.

$1.05^{14}\approx 1.98<2$, but $1.05^{15}\approx 2.08>2$; so in week $15$ you pass double.

$s_{14}\approx 1.98<2<2.08\approx s_{15}$

But do not stop at week three: there you are only about sixteen in a hundred better. The longer you go on, the steeper it gets: repeated multiplying by 1.05 ends up beyond every number. Practise a little, every week, and write to me when you pass double.

In week $3$ it is only $1.05^3\approx 1.16$. But the longer you go on, the more it grows: ${\lim }_{n\to \infty }1.05^n=\infty$.

$s_3\approx 1.16$, ${\lim }_{n\to \infty }{s}_n=\infty$