An old billionaire is writing his will, and deciding how to distribute his nine estates among his three children. he wants to give each child at least one estate, and he does not want to give any two of his children the same number of estates. in how many possible ways can he distribute his estates?

18

Step-by-step explanation:

6 2 1    How many ways does this produce and are they legitimate?

So any combination that you come up with, will produce 6 variations within those three choices. They are legitimate.

Lets see what you get with at most 5 estates to one person

5 3 1    Which will also give 6 variations. There are no other ways to make 5

4 3 2   Which will give 6 variations

I don't think there are any more possible choices.

Are the only 3 ways the estates can be split given the rules. Each choice leads to 6 other choices. The total is 18

geometric series converge the common ratio r has magnitude less than one.

which is equivalent to

for the example we need

|2-x| < 1

-1 < 2-x   and 2-x < 1

x < 3 and x > 1

we can write that

answer: 1 < x < 3

1a. common ratio r=250/25=10 > 1     diverges

1b.   r=50/100=1/2 converges

when these converge they converge to

where a is the first term.

answer: 200

1c. r=-1   so it's not the case that |r| < 1.     diverges

2a. x is the common ratio so we converge when |x|< 1

2b.   r=-10x / -2 = 5x.

we converge when

|5x| < 1

|x| < 1/5

2c.   r=3-2x

-1 < 3-2x and 3-2x < 1

-4 < -2x and 2 < 2x

x < 2 and 1 < x

1 < x < 2

2d. [/tex] r=-48x^3/12 = 4x^3[/tex]

|4x^3| < 1

|x^3| < \frac 1 4

|x| < 4^{-1/3}