Weird little regularity

[alkatran]

Take 1$ (in ANY base: 10, 8, 2...)
Now, assume that interest is 10%

write down the numbers for the years 0 through 4
Now write down the nCr pyramid next to it (pascales triangle?)

See any resemblance?

1
1.1
1.21
1.331

   1
  1 1
 1 2 1
1 3 3 1

Oh, by the way, I know why it works.

[Darkwraith]

Why does it work?

Also, is it compounded or is it simple?

[riis]

Originally posted by Darkwraith
Why does it work?

Well, if you multiply something with 11 (I know Alkatran used 1.1 in his example, but explanation is easier with 11), you're in fact adding the tenfold of the number with the number itself. To multiply a number with 10 (the radix of the decimal system!), all digits are shifted one position to the left, and a zero is added to the right. This number is added to the original value, so the "target" digit has the same value as the "source" digit and the one to the left of it.

The Pascal triangle is generated by adding numbers as well. A new number is calculated by adding the number to the upper left (digit in number which got multiplied by 10) and the number to the upper right (digit in the original number).

I hope this was understandable

[riis]

Oh, and in Alkatran's example (1.1) the number is shifted to the right, causing the same effect, but the resulting number is ten times lower.

[Darkwraith]

How are you doing interest?

[alkatran]

Simple formula for interest that I used:

T(0) = 1
T(x) = T(x-1)*10%

or, T(x) = 1*(1.1)^x


A pascale's triangle is generated by adding the two number above a number. (for example, 1 and 1 makes 2, as seen in the third row, center). When you multiply something by 110%, you are adding that number, and a tenth of that number. So like this:

1 +
0.1 = 1.1
1.1 +
0.11 = 1.21

The same basic thing is used in both cases.

[sql_lall]

Same thing happens with powers of 11:
11, 121, 1331, 14641,....WHOOPS

Of course, you screw up when you get to the next one, cos of course 1 5 10 10 5 1, the '10's have two digits.

you can carry over though, to get 161051!

It's easy to reason, as if X is ont number, then
11 X = 10X + X...so each digit is the sum of the two adjacent digits in the previous sum...like pascal's triangle, as mentioned before.

That's also why it stuffs up past 5

[alkatran]

Of course the digits carry over. The regularity is kept, it's just less obvious in lower bases.