1)how many ways can the letters in the word bookkeeper be arranged? (must show set-up )
2)seven swimmers compete in the championship freestyle 50m race. how many ways can first, second, and third place prizes be awarded? you must show your set-up
3) katie has a bag of candy consisting of 9 red gumdrops, 15 orange gumdrops, and 7 green gumdrops. if she eats 2 of the gumdrops one after the other, what is the probability that the following will occur? reduce all fractions.
a) the first candy eaten was orange and the second one was red:
b) both gumdrops were green in color:

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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}