Mr. Hammond, I should have listened.
November 12, 2014 3:17 PM   Subscribe

I want to calculate a sequence. It's built on some kind of exponential magic. That is as much as I know. Please hope me.

Said sequence is here. Repeated below:
1) 4.0, 4.7, 5.6, 6.7, 8.0, 9.5, 11.3, 13.5, 16.0...

2) 7.5, 8.9, 10.6, 12.6, 15.0, 17.8, 21.2, 25.2, 30.0...
I threw some numbers at the screen and I thought I had it solved:

4.00**0.125 == 4.75
4.75**0.125 == 5.65
5.65**0.125 == 6.72
...

but that doesn't work with the sequence that starts with 7.5.

I understand that we have: 4, 8, 16, 32 and 7.5, 15, 30, 60. But how are the 1/4 intervals calculated?

My brain hurts. Please, could someone hold my hand?
posted by popcassady to Science & Nature (6 answers total) 1 user marked this as a favorite
 
Each f-stop is a doubling. In those pictures, each f-stop is a red number. Note, they each double.

The numbers in between are fractional powers of 2. So, the sequence is:

7.5 × 20.00 = 7.5
7.5 × 20.25 = 8.9
7.5 × 20.50 = 10.6
7.5 × 20.75 = 12.6
7.5 × 21.00 = 15.0 (first f-stop after 7.5)
7.5 × 21.25 = 17.8
7.5 × 21.50 = 21.2
7.5 × 21.75 = 25.2
7.5 × 22.00 = 30.0 (second f-stop after 7.5)
7.5 × 22.25 = 35.7
7.5 × 22.50 = 42.4
7.5 × 22.75 = 50.5
posted by saeculorum at 3:28 PM on November 12, 2014


It's a geometric sequence, where every 4 steps it doubles. That means every single step is multiplied by the fourth-root of 2. That winds up being close to 1.189.

The second sequence has the same factor each step, but it starts out 1.875 larger, and stays that way.
posted by aubilenon at 3:29 PM on November 12, 2014


Best answer: Just to be a bit more general:

A geometric series has a fixed coefficient (4 or 7.5) and a common ratio between terms. You can calculate the ratio by just dividing one term by the preceding one. However in this instance, I didn't do that - I noticed that there's an integer ratio every 4 steps. If you want to multiply by something four times to double it, that'll be the 4th root of 2.
posted by aubilenon at 3:41 PM on November 12, 2014 [1 favorite]


Best answer: If you're guaranteed that the sequence is an exponential function, then you can approach this in general as follows:

generically, exponential functions look like

f(t) = A b^t

where A is some constant and b (the base of the exponential function) is greater than 0.

So, for the first sequence (counting terms as 0, 1, 2, 3, ...), we know that f(0) = 4. But we also know that f(0) = A*b^0= A. So hooray, A = 4.

So now we know that f(t) = 4*b^t.

But you also know that f(4) =8, since the fourth term in your sequence is 8. But using the function definition, f(1) = 4 b^4. Therefore, 8 = 4b^4, so b^4 = 1/2, so b = 2. Therefore, b = 2^(1/4).

(Notice also that your sequence terms are rounded---I originally was going to use f(1) = 4.7, so b = b^1 = 4.7/4 = 1.175, while 2^(1/4) = 1.189207115, which is a pretty big difference.)

The nice thing about this technique is that it's completely algorithmic: if you want to determine an exponential function that goes through a couple of points, it's easy to do so using the fact that you know what the generic function looks like, either f(t) = Ab^t or f(t) = A e^(kt) (where e is Euler's constant ≈2.71828..., the base of the natural logarithm), which are two different ways to represent exponential functions.

If we do the same analysis to the second function, we see that f(4) = 15, and f(8) = 30, so Ab^4 = 15 and Ab^8 = 30. (I'm avoiding the decimals, because I don't trust them to be accurate, see above.) Dividing the second equation by the first, we get that

(Ab^8)/(Ab^4) = 30/15 ==>
b^4 = 2

Oh, look, b = 2^(1/4) again. And this time f(0) = 7.5 --> A = 7.5, so f(t) = 7.5*(2^(1/4))^t

Now you can use this function to calculate any number of stops. You want to know what f(20) is for the second sequence? (that is, the 21st entry in the sequence, because I'm counting 0,1,2,3,...)

f(20) = 7.5*(2^(1/4))^20 = 7.5*2^(20/4) = 7.5*2^5 = 7.5*32 = (15/2)*32 = 15*16 = 240.
posted by leahwrenn at 3:49 PM on November 12, 2014


Response by poster: Jesus.

Thank you. Problem solved! You people are too clever.
posted by popcassady at 3:54 PM on November 12, 2014


More simply, the first one (in terms Google Calculator will accept) is
(sqrt sqrt 2) ^ 8
(sqrt sqrt 2) ^ 9
(sqrt sqrt 2) ^ 10
etc.
posted by w0mbat at 7:36 PM on November 12, 2014


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