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There is some complicated (for me) math involved with a game I've been running, set on a Dyson Shell in the process of being built. These questions would take me a long time to answer, my math skills being abysmal. I don't know if there's a fast way to answer them, but perhaps someone here does? If some kind saint can and will answer them I will be overflowing with gratitude.

Here's the premise:
1000 layers form over 1000 years, the formation rate accelerating over time. Layer 2 forms 1% faster than layer 1, layer 3 forms 1% faster than layer 2, and so on out to the 1000th layer.

And here are the questions:
How long did layer 1 take to form?
How far into the 1000 years did layer 500 form?
How long did layer 1000 take to form?
How long from the beginning until layer 1414 forms?
How long from the beginning until layer 23452 forms?

Basically, you have an inverse compounding situation. So instead of X*1.01, you have X/1.01.

The first thing you need to do is define how long a year is.

Take that value and divide by 1.01 to get the value for the next layer. Write down the result and repeat as needed.

If you have the skills, you could probably code an Excel to do this automatically--just remember to include a SUM to give you the total time for any layer.

If layer 1 takes X years to form, then
layer n takes (.99) (to the n-1)X years to form

The sum of the first 1000 terms (ie how long it takes 1000 layers to form) can be calculated using the formula at
http://en.wikipedia.org/wiki/Geometric_series#Formula with a=X, r=0.99 and n=1000 as


1-(0.99 to the 1000)
X-------------------------
1-0.99

=X times 99.999956828752589341749011367171

Now, you have said that that =1000 years

so X is 1000 divided by 99.999956828752589341749011367171 years

or 10.000004317126604823232698451593 years

Your remaining questions are either of the form how long did layer N take to form, which now we know is (0.99 to the (N-1)) times 10.000004317126604823232698451593 years

Or how long did it take total until layer N had formed which is

or (1-(0.99 to the (N-1)th) times 10.000004317126604823232698451593 years divided by (1-0.99)

or (1-(0.99 to the (N-1)th) times 1000.0004317126604823232698451593
For large values of N, (0.99 to the (N-1)th is very small. Indeed the series that gives the sum of the time it takes to form the first N layers converges to 1000.0004317126604823232698451593 years or a spot under 3 hours 47 minutes over the 1000 years.

Thanks Keldorn. You've left me with a sore brain, but perhaps using this thingmabob http://web2.0calc.com/, I can make some sense of it. Not that I know what half the buttons do.

So if I understand you correctly the answer to:
How long did layer 1 take to form?
is: 10.000004317126604823232698451593 Right?

And to get my answer to:
How far into the 1000 years did layer 500 form?
I need to enter:
(1-(0.99 to the (499)th) x10.000004317126604823232698451593/(1-0.99)
Which goes into the calculator as:
(1-(.99^499))*10.000004317126604823232698451593/(1-0.99)
Giving me the result of:
993.36357728945306028154056127980541 Right?

And to get my answer to:
How long did layer 1000 take to form?
I input:
(.99^999)*10.000004317126604823232698451593
Getting me the answer:
0.0004360733944265891614597568514
Or as about 3.8 earthly hours.

Answering:
How long from the beginning until layer 1414 forms?
I input:
(1-(.99^1413))*10.000004317126604823232698451593/(1-0.99)
Getting:
999.9997516794012149059957876293692
(ok I know that's not quite right as it's under the original 1000 year period)

Answering:
How long from the beginning until layer 23452 forms?
I Input:
(1-(.99^23451))*10.000004317126604823232698451593/(1-0.99)
Netting me:
1000.0004317126604823232698451593

What these results tell me, especially the last few is that I need to flatten those results a bit And perhaps that these calculations are rounding in the calculator more than they should.

Shortly I'll try to get my head around the bit of math you gave me at the start. Perhaps try replacing the 0.99 ramping with 0.999 or whatever ends up working. But first this is the "Gaming Discussions" section so I'll explain why I've sought these screwy numbers, but I'll start a fresh post for that as it might take me awhile....

At the end of last year I thought up a new system (which would require multiple posts of its own to explain so I won't go into that). My weekly gaming group were breaking for Christmas, and with a player dropping out for work reasons we wouldn't be continuing my then current D&D3.5 game when we resumed in the new year. I offered a Pathfinder campaign option, Mage the Ascension following up on an old campaign, or trying out the new system in a very different sci-fi/modern/fantasy setting running on a partial Dyson Shell. The players chose the new system so I scrambled in the next few weeks to polish the system and flesh out the setting.

More details on Dyson Shells can be found by looking up Dyson Spheres on Google, but put simply it's equivalent to 550 million earths in surface area. Why did I want to use one? I'm a big fan of the writer Larry Niven, best known for his Ringworld series. The Ringworld is a Dyson Strip, a ribbon wrapped around a sun, with an enourmous livable surface inside spun up to a near Earth gravity. The series established that the builders didn't have artificial gravity, nano technology, or AI's. But by the time of the Ringworld stories there were other races that did have those. So with some willingness to go beyond 'science fiction' and more into 'science fantasy' I went bigger. I also roughly borrowed Robert A Heinlein's multiverse theory from his book "The Number of the Beast", and have since ripped heavily from The Martian Tales of Edgar Rice Burroughs.

What the players landed on was a Dyson Shell still being built, walled up into map like copies of preexisting worlds. The shell has been expanding for roughly a thousand years, and they found out later that there are roughly a million other walled worlds beyond that one. This is why the thousand years and thousand layers were the baseline I gave out for my math. The characters haven't travelled much since the game begun so most of my focus has been on their immediate surroundings. There is a curve as the shell is growing around a sun, but one thing I found out early on is that between the high walls and the visual distortion of looking sideways through a lot of atmosphere, visibility doesn't go far. If worlds were curved high enough they'd be visible above, but as they started on an older world near the core, and less than 1% of the shell was complete, nothing was above as yet.

Recently we took a break so someone else can run a game. At the end of the last session I managed to get them teleported nearer to new layers (quite a cliffhanger when you consider they haven't gone outside the starting continent in around 20 in-game years).

So now I really should cement some things. How fast are new worlds forming? When will the shell finish? When will it double from the games starting million? How fast did worlds form at the start? Those are the questions that went into the math I begged for help with. The 1414 question will answer when the shell 'fragments' get to 2 million, and the 23452 question will give me a finish date.

I could just make up numbers, but given the option I'd prefer to be a bit cleaner about it. I know the creator is a powerful AI of earth origin. Anyone who's read Niven's recent "..of Worlds" books (Ringworld prequels) should know what I mean by calling it a souped up Jeeves. One of the reasons Jeeves is building the shell is to protect copies of various life from the galactic core explosion established in various Niven books, which will eventually sterilize much of the Milkyway Galaxy. There's no rush to finish the entire shell, so it makes sense (to me) for Jeeves to set its growth to a mathematical schedule.

There's much I've skipped describing. Player observed science marvels such as artificial suns, varied gravities, dormant shadow squares, large red sun being pulled apart for building materiel, the scrith walls and floor materiel, and more. There's also the matter of non Niven's 'Knownspace' worlds being added to the mix. And there's the memory loss and nanotech all the player-characters started with. If anyone finds any of this interesting I can go into greater details, but for now I'll leave off and try puzzling out that math.

Wasywha? I'm getting lost on some of the math, not sure how that first stuff goes in the calculator, and now the numbers are just running together to my eyes. Too late at night, I'll have to try puzzling this stuff out with a fresher head. Plus I think I need to find a calculator that doesn't round so roughly.

Do you have a copy of BIll Coffin's Septimus setting? It's set inside a Dyson sphere, and might give you some more ideas if you need to suddenly create another area.