I'm trying to find the Rate of decrease at which 1,262 is multiplied by every day (16 times in total) and added to the Base number of 17,584 in total 16 times to get to the goal number of 24,000. I've started to work out a formula but I know I'm way off.
$24,000=17,584+(1,262\times R)^{16}$
Here's my question phrased as an example: I've spent 17,584 ℎ ℎ. , 1,262. I need to lower my daily spend by R each day for the next 16 days to end up spending $24,000 total.
What you are effectively looking for, is a geometric series with $t_0=1262$, and $r$ such that $t_{16}=24000 - 17584=6416$. There are several ways to do this: use the initial formula for a geometric series and solve for $r$, or use one of the rearranged formulae to obtain $r$ directly.
If the first of the remaining sixteen days is allowed 1262 currency expenditure as well, you can adjust this.
Your equation will not work for this purpose. You are instead calculating something that makes no sense in this context.
If you instead removed the brackets, you would be finding an $R$ that provides no further information than what rate to decrease the initial spending by, each day, so that you could spend all of the remaining money on the last day only (and not daily as you need).
A brief check with Excel gives back $r=0.845$ as getting you within a single unit's accuracy, which for real expenditure, is as good as you are likely to need. That is, reduce each day's spending by 15.5% compared to the day before.