# Limit when recursively adding percentages

Suppose that I wanted to recursively add a percentage to some number. For example:

20,000 + 10% = 22,000

22,000 + 10% = 22,200

22,200 + 10% = 22,220

etc.

we can easily see the limit is 20,000*(10/9). I want to know how I arrived at this. What is the general rule? For example, suppose we started with 42,000 and applied 27% recursively. How would I find the limit? Thanks.

• Ever heard about geometric series? – Vincent Aug 31 '16 at 14:53
• yes, but I'm not sure how to apply it to this – Sina Madani Aug 31 '16 at 14:56
• Your number would stand for the first term ($u_0$) and your percentage for the coefficient $c$ where $u_{n+1}=c\times u_n$ (expressed as a real, for instance 10% gives $c=0.1$) – Vincent Aug 31 '16 at 15:01

that is

$i + i^2 + i^3 + i^4....$

multiply by $(i - 1) / (i - 1)$

$[i + i^2 + i^3 + i^4....](i - 1) / (i - 1)$

$= [(i^2 - i) + (i^3 - i^2) + (i^4 - i^3) + (...) + ...] / (i - 1)$

cancel similar terms, only i remains on the top, all others cancel indefinitely

$= -i / (i - 1)$

$= i / (1 - i)$

for 10%, your modified interest rate becomes $0.1 / (1 - 0.1) = .111111 = 11.111..$%

Nevermind, just realized I could use the following rule:

a/(1-r), where a = starting number, r = the percentage, to find the limit.

e.g. 20000/0.9 = 22,222...

• similar to mine with 1 + i / (1 - i) = 1 / (1 - i) – Cato Aug 31 '16 at 15:07

By the way, "20,000+ 10%= 22,000" makes no sense. What you mean is "20,000+ 10% of 20,000= 22,000" which is the same as "20,000+ 0.10*20,000= 1.10(20,000)".

• percentages are always relative though, I assumed that's what it meant. If you try it on a calculator, it works. – Sina Madani Aug 31 '16 at 15:15