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Here's the task at hand:

  • I have the first and last values (a range from 0 to 100)
  • I have the number of steps (20)
  • I need to find values for all 20 steps

So, a simple arithmetic progression would be to use the difference of 5 for each step. However, I want the difference between steps 19-20 to be the biggest and the difference between steps 1-2 to be the smallest, with the rest of values progressively between the two.

What are my options? How do I go about solving this?

artooras
  • 101

1 Answers1

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A geometric progression is not possible between 0 and a non-zero value.

You could use a power function instead of an exponential function (which is what a geometric progression would give). Pick a $p > 0$, not necessarily an integer, and try a function of step index to value like:

$$ f(t) = 0 + (100 - 0) \left(\frac{t}{20}\right)^p $$

$p = 1$ corresponds to arithmetic progression.

[edited to add]

Perhaps you want the step sizes to be in geometric progression, rather than the values themselves? In which case, write $a$ for the first step size, and $r$ for the ratio between them, so step $k$ is $a r^k$, then the sum is:

$$ 100 = a \frac{1 - r^{20}}{1 - r} $$

Solving for $a$ given $r$ is trivial, solving for $r$ given $a$ could be done using numerical root finding methods.

Claude
  • 5,647