# Thread: Compressing a curve/sequence of numbers

1. ## Compressing a curve/sequence of numbers

[Apologies if this is in the wrong sub-forum as my maths knowledge is not much more than basic]

Here is a sequence of numbers, starting at 1 and then doubling at each step until 128 is reached. It would produce a nice curve if plotted on a graph.

I - 1
II - 2
III - 4
IV - 8
V - 16
VI - 32
VI - 64
VIII - 128

If I wanted to reduce the sequence of numbers from eight to five, while retaining the curve and keeping the first and final numbers of the sequence the same, how would I calculate this?

So compressing it like this:

I - 1
II - ?
III - ?
IV - ?
V - 128

Also, while I am here, how would one calculate stretching the sequence in the perpendicular direction? So keeping a sequence of eight numbers this time, with 1 remaining as 1 and the final number (VIII) being diminished from 128 to, for instance, 100 and the curve adjusting relative to it?

So like this:

I - 1
II - ?
III - ?
IV - ?
V - ?
VI - ?
VI - ?
VIII - 100

2. ## Re: Compressing a curve/sequence of numbers

$a_n = 2^n,~n=0,1,2,\dots$

now you want to make a new sequence $b$ such that

$b_n=1,~b_5=128$

ok, let's assume the same form as the first sequence just with a different base

$b_n = \beta^n$

$b_0 = \beta^0 = 1$ so we're good there

$b_5 = \beta^5 = 128$

$\beta = (128)^{1/5} \approx 2.64$

$b_n = \left((128)^{1/5}\right)^n = (128)^{n/5}$

same deal for your 2nd problem

$c_n = \gamma^n$

$c_5 = \gamma^5 = 100$

$\gamma = (100)^{1/5}$

$c_n = (100)^{n/5}$