A mathematical question for no real reason!

[Stew]

Ok, so bear with me on the description on this as it may not make sense.


This number sequence:







The next value = the previous value + the previous difference to add multiplied by three.


Can anyone write a formula on how to calculate, for example, the 15th value without having to calculate the previous 14? I have been pondering and my meager brain can't do it.


There is no real end goal on this - it's just curiosity.


It may not be possible, of course.....

 

[nic a char]

Intriguing Stew, but sorry, I haven't a clue!

 

[spandit]

If place number = x and value = y then:

Hang on... made a cock up in the formula...

The 15th value (x=15) is 1,594,323

 

[Stew]

If place number = x and value = y then:

y=(x-1)+(3^(x-1))

So, if x=6 (i.e. place number 6), y=122 (41+81)

The 15th value (x=15) is 1,594,323

Not that I can see using excel.

I've had it suggested that the first difference should be 3 not 1 but if the first difference is 1, second 3 (as above) then the 15th value = 2391485. If the first difference is 3, second is 9 then the 15th value = 7174453. That's using two lines of excel formulas.

Forgive my ignorance - what's the up arrow signify after the 3 in your formula?

 

[Stew]

Aaah, now then! 1594323 is the difference between 14 and 15 of that first sequence!

Seems like a good step in the right direction!

 

[spandit]

Sorry, made a complete hash in the spreadsheet I was using to calculate the value.

15th value, I am now pretty sure, is now 2,391,485 but it's summing powers of 3 (the up arrow symbol means "to the power" so 3^3 is the same as 3*3*3 or 3 cubed etc.)

 

[Stew]

Well you have the right answer - what's your formula?

 

[spandit]

Well you have the right answer - what's your formula?

I'd initially calculated that by using a spreadsheet and adding numbers as I went along, much like you probably did. However, I wasn't satisfied that a formula couldn't be found so after a bit of searching around, I wrote this:

y=2+(3*(1-(3^(x-2)))/-2)

So, for example, if x=4, then y=2+(3*(1-(3^2)))/-2; y=2+(3*(1-9))/-2; y=2+(-24/-2); y=2+12; y=14

Simple

 

[spandit]

The numbers start getting pretty huge. For example, by the time x=30, y=34,315,188,682,442

 

[Stew]

Check you out.

Thankyou!!!

 

[spandit]

No problem, although it was harder than I'd originally thought. Actually, I think the formula could be simpler...

y=1+(1-3^(x-1))/-2

Got the formulas from here:

http://mathworld.wolfram.com/GeometricSeries.html

It's essentially formula (6) but adding 1 because you started the series from there...