Decreasing power?

Question

Say you have numbers X and Y

When Y is 1 you want to times X by .5,

When Y is 2 you want to times X by .5 then by .25,

When Y is 3 you want to times X by .5, then by .25 then by .125, and so on.

How can I do that in a spreadsheet? It's like a ^ but it halves itself every step?

score 1 · Answer 1 · answered Aug 12 '13 at 11:22

1

What you need to get this effect is to do the following:

Let $Z = \prod_{i=1}^Y 2^i$ so what you want to do is multiply $X$ by $Z^{-1}$.

Now, we get that $$Z = 2^{\sum_{i=1}^Y i} = 2^{\frac{Y(Y+1)}{2}}$$

So this is the number you need to multiply by the inverse of.

answered Aug 12 '13 at 11:22

Tobias Kildetoft

21,247

Thanks. So would I write that in a spreadsheet as: =X(2(y*(Y+1))/2)? – skybreaker Aug 12 '13 at 11:38
@skybreaker How to write in in a spreadsheet depends on the syntax used in that spreadsheet. – Tobias Kildetoft Aug 12 '13 at 11:39
I just mean a regular google spreadsheet. – skybreaker Aug 12 '13 at 11:40
@skybreaker How was anyone supposed to guess that from the word spreadsheet? Anyway, I have no idea what the syntax is there. You should probably look it up in the documentation. – Tobias Kildetoft Aug 12 '13 at 11:41
ok, sorry. thanks for your help. I don't know which part of your formula I'm supposed to be converting. But I'm trying ^_^ – skybreaker Aug 12 '13 at 11:49

Riccardo Del Monte · Answer 2 · 2013-08-12T16:21:40.340

1

Seems you'd like a function $x = f(y)$.

As you are saying, when the variable $y$ increases by one, the function $x$ has the same values for $y-1$, plus the last term multiplied by $\frac{1}{2}$. To better understand, you are dealing with the geometric series

$$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$$

that is

$$\sum_{n=0}^{\infty} \frac{1}{2}\frac{1}{2^n} $$

Now, since a closed form for a geometric series is $a\frac{1-r^n}{1-r}$, and since in this case we have $a = r = \frac{1}{2}$, the function $x$ can be defined as

$$x = \frac{1}{2} \frac{1-\frac{1}{2^y}}{\frac{1}{2}} = 1-\frac{1}{2^y}$$

edited Aug 12 '13 at 16:21

answered Aug 12 '13 at 11:42

Riccardo Del Monte

517

Thanks. I have no clue how to convert this to spreadsheet language. I guess I'm out of my depth. I will search for a spreadsheet-orientated stackexchange... if one exists. – skybreaker Aug 12 '13 at 11:52
Where did you get any sort of division by powers of $5$ here? $.5$ is not the same as $\frac{1}{5}$. – Tobias Kildetoft Aug 12 '13 at 12:59
@TobiasKildetoft Oh my goodness!!!!! I just... got confused! Sorry :) Now I've already edited the post – Riccardo Del Monte Aug 12 '13 at 16:22

Decreasing power?

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