A few math problems

Just a few problems I came up with, none of them too hard.

1). 1,2,5,14: find the next term and a formula for the nth term.

2). I have 7 dice, each of which I role 1 time per game. I have to pay $0.50 per game. If the total sum of all my dice is less than 10 or more that 39, I win the jackpot of $1,000,000. Is this game worth playing, assuming I have all the time in the world, will continue playing for ever? What is the expected gain/loss for every time I play?

3). Not really a problem, just something to think about: Is infinity + 1 greater than or equal to infinity?

Originally posted by clandarkfire
Just a few problems I came up with, none of them too hard.

1). 1,2,5,14: find the next term and a formula for the nth term.

2). I have 7 dice, each of which I role 1 time per game. I have to pay $0.50 per game. If the total sum of all my dice is less than 11 or more that 38, I win the jackpot of $1,000,000. Is this game worth playing, assumin ...[text shortened]... lly a problem, just something to think about: Is infinity + 1 greater than or equal to infinity?

2) Hell yes you play. The odds of 7d6 being being greater than 39 or less than 10 are pretty good. I simulated 10k trials and won 3 rolls. In fact, this is even a good game if you have to roll either a 7 or a 42, because the odds of rolling either are about 7 in a million, so after a million trials you'd expect to be up about $6.5 Mil. That's a pretty good player advantage.

3) There are only two magnitudes of infinity. Countably infinite, and uncountably infinite.

An example of "countably infinite" is the number of integers, the number of non-negative integers, or the number of even integers. It's easily proven that the magnitude of each of these things are the same.

"Uncountably infinite" would be like the number of real numbers, the number of fractions between 0 and 1, or the number of irrational numbers.

Uncountably infinite is larger magnitude than countably infinite

*Edit*, messed up the bounds on the dice game, re-calculated my sim.

Originally posted by forkedknight
3) There are only two magnitudes of infinity. Countably infinite, and uncountably infinite.

An example of "countably infinite" is the number of integers, the number of non-negative integers, or the number of even integers. It's easily proven that the magnitude of each of these things are the same.

"Uncountably infinite" would be like the number ...[text shortened]... countably infinite

*Edit*, messed up the bounds on the dice game, re-calculated my sim.

Fractional numbers, of the type p/q where p and q are integers, and q is not zero, are countable as shown by George Cantor. Real numbers are not.

As infinity is not a number, normal laws of arithmetics doesn't apply. But it can be shown that inf+1 is exactly equal to inf itself. Also shown by Cantor.

Originally posted by FabianFnas
Fractional numbers, of the type p/q where p and q are integers, and q is not zero, are countable as shown by George Cantor. Real numbers are not.

As infinity is not a number, normal laws of arithmetics doesn't apply. But it can be shown that inf+1 is exactly equal to inf itself. Also shown by Cantor.

I think that by "fractions" he meant connected intervals of [0,1]. Or maybe not...

Originally posted by Palynka
I think that by "fractions" he meant connected intervals of [0,1]. Or maybe not...

If he didn't meant fractions as in http://en.wikipedia.org/wiki/Fraction_(mathematics), then he'd used the same formulation that you did.
Or maybe not...

41 but im still looking for a formula

Originally posted by FabianFnas
If he didn't meant fractions as in http://en.wikipedia.org/wiki/Fraction_(mathematics), then he'd used the same formulation that you did.
Or maybe not...

I meant exactly what you thought I meant, and I was wrong. The rest of my examples stand.

Oh, and for 3) a recursive definition for the nth term of your series is S(0) = 1, S(n) = S(n-1) * 3 - 1

1)

1, 2, 5, 14, 42

Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!).

2)

There are 6^7 = 279936 possible results.
Of those there are 36 results in which the sum is less than 10.
There are also 36 results in which the sum is more than 39.
Chance of winning is therefore 72/279936 = 1/3888
Expected profit per game: 1/3888 * 1,000,000 + 3887/3888* 0.50 = 257.70
Yes, you should play this game. A lot.

3)

Depends on what subject. In general infinity is not a number, so infinity + 1 does not have any meaning.

In Axiomatic Settheory (for example), Omega represents the 'number' that comes after al the natural numbers. Then, Omega + 1 is larger.

Originally posted by TheMaster37
1)

1, 2, 5, 14, 42

Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!).

2)

There are 6^7 = 279936 possible results.
Of those there are 36 results in which the sum is less than 10.
There are also 36 results in which the sum is more than 39.
Chance of winning is therefore 72/279936 = 1/3888
Expected profit per game: 1/3888 * 1 ...[text shortened]... ga represents the 'number' that comes after al the natural numbers. Then, Omega + 1 is larger.

Thank you for using the Online Encyclopedia of Integer Sequences.

3) could also be
1,2,5,14,41

With S(n) = (3^n + 1)/2

Originally posted by forkedknight
Thank you for using the Online Encyclopedia of Integer Sequences.

3) could also be
1,2,5,14,41

With S(n) = (3^n + 1)/2

😉

Originally posted by forkedknight
I meant exactly what you thought I meant, and I was wrong. The rest of my examples stand.

It was just a correction, not a critic.

inf+1 = inf
inf*2 = inf
inf*inf = inf
inf^inf = inf
as long as inf = the number of integers, meaning countable infinity.

inf/inf and inf-inf however lacks meaning.

OMG the answer really was 42?!

Originally posted by AThousandYoung
OMG the answer really was 42?!

In theory, 42 would work. I was looking for 41 though...

Originally posted by clandarkfire
In theory, 42 would work. I was looking for 41 though...

You are nothing to the Second Smartest Computer That Ever Was.

Originally posted by FabianFnas
It was just a correction, not a critic.

inf+1 = inf
inf*2 = inf
inf*inf = inf
inf^inf = inf
as long as inf = the number of integers, meaning countable infinity.

inf/inf and inf-inf however lacks meaning.

I really meant that as more of a response to Palynka.