Expert: Abe Mantell - 5/24/2007

Question
Dear Mr.Mantell,
My problem is:
I would like to be able to produce a list of numbers derived exponentially, where the sum of this list is equal to a value that I decide on. Specifically:
if Total Time = 264 seconds
Number of Steps = 198
Initial Step = 0.167 seconds (equivalent to 90 beats per minute)
Final Step = 12 seconds

I can't seem to vary my method to fit the Total Time!

Here's what I do:
12/0.167 = 71.856
then 71.856 to the power of 1/197 = 1.0219359
then 0.167 * 1.0219359 = 0.171 (second step)
2nd step * 1.0219359 = 3rd step
and so on. I certainly reach the golden value of 12 seconds after a while (!!!) but, on adding it all up it very quickly shoots over my 264 second boundary.
I would be really grateful if you had a solution here, and especially one that I could apply to other values.

Thankyou very much for your time and consideration,


Yours sincerely,

Alistair

Answer
Hello Alistair,

I think you want to use the sum of a geometric progression.
A+Ar+Ar^2+...+Ar^n = A[1-r^(n+1)]/[1-r]

For example, your first case: A=0.167, n=198 (or 197??)
and you want the sum to be 264...now we solve for r:
264=0.167[1-r^199]/(1-r)...which gives r=1.016806275
But then the final step is 4.52776 seconds, not 12.
If we get the final step to be 12, then the sum will
not be 264...unless we start with a different initial step
or change the number of steps.

Abe