### 1.

Given a growth rate and starting value, how many iters it takes to reach a certain value?problem: start * pow(grow, iters) = end. find iters. solution: pow(grow, iters) = end / start. divide by start. iters * log(grow) = log(end / start). take natural logs. iters = log(end / start) / log(grow). divide by log(grow) iters might not be an integer value!

### B.

Given a growth rate and starting value, how many iters it takes for the sum of the results to reach a certain value?..I just now figured this formula out experimentally.. bombs out if grow=1 sum(i=1,iters. of pow(grow,i)) = (pow(grow,iters) - 1) / (grow - 1) problem: start * ((pow(grow,iters) - 1) / (grow - 1)) = sum. find iters. solution: (pow(grow,iters) - 1) / (grow - 1) = sum / start. divide by start. pow(grow,iters) - 1 = (grow - 1) * sum / start. mult by (grow - 1) pow(grow,iters) = 1 + (grow - 1) * sum / start. add 1. iters * log(grow) = log(1 + (grow - 1) * sum / start). take natural log. iters = log(1 + (grow - 1) * sum / start) / log(grow). divide by log(grow). iters might not be an integer value!

### III

Given a certain result value and a number of iterations, what should the growth rate be?problem: start * pow(grow, iters) = end. find grow. solution: pow(grow, iters) = end / start. divide by start. iters * log(grow) = log(end / start). take natural logs. log(grow) = log(end / start) / iters. divide by iters. grow = exp(log(end / start) / iters). exponentiate.

### Fourth:

Given a certain sum of the results and a number of iterations, what should the growth rate be?problem: start * ((pow(grow,iters) - 1) / (grow - 1)) = sum. find grow. solution: (pow(grow,iters) - 1) / (grow - 1) = sum / start. divide by start.

hmm this one gets more involved.. I'll leave it for now..

- — james mccartney (quoted from a mail)