n=n+1%=?

 Maths Puzzle 

“Every day in every way I get better and better.”

“Even if you improve by only 1% a day, that is progress.”

So if I improve by 1% a day, how long will it take for me to be 100% better? 

It’s not 100 days because we value it based on the previous day, not based on day zero. Only day 1 is based on day zero. Day 2 is based on day 1’s value. Day 3 is based on day 2’s value. And so on. 

How many days does it take to achieve 100% improvement? 

There is a formula but I can’t work out what that formula is. 

Something like:

n = value 

n + (n x 1%) until n=100% = ?

But ‘until’ is not a proper maths formula.

I got as far as;

n = n + (n x 1% )

Although this formula does not separate it into incremental improvements over however many days. 

I know from computer coding that were I to ask a computer to do a math such as this it will jam the machine up for ever. The code would need a stop button to quit the function. 

So this formula is not the solution to the question. It is a step toward understanding it. 

We need to differentiate yesterdays n to todays n because todays n is 1% greater than yesterdays, by however much value that is. 

The lengthy manual way to do this is to repeatedly keep applying the basic formula 

1% of yesterdays n + yesterdays n = todays n 

until we achieve n = 100% or as close to it as possible. 

A friend asked me what the n for day one is. 

I had assumed it to be 1 as in 1% for simplicity and elegance to make all the other number-crunching congruent with the concept that 100% = 100, although it does bare taking awareness the stating number could be anything. 

This is more important if anyone ever has recourse to do anything practical with these formula. 

For example, working out the escape velocity and traction gain against atmospheric friction in escaping planetary density into space. Both gravitational and in terms of time-dilation. 

Things we would assume space agency scientists would have figured out a long time ago. 

It’s literally rocket science so don’t feel disheartened if you can’t do it either. No nation wants to hand its enemies the solution to building surface-to-air missiles on a plate. 

Perhaps why the answer to this original question is not easily available in public knowledge.