There are 5 green, 1 red, and 1 blue book on the shelf. how many ways can they be arranged if the red and blue book are separated?

3,600

Step-by-step explanation:

There are 5 green, 1 red, and 1 blue book on the shelf, 7 books in total.

First, count the number of ways when the red and the blue books are not separated. If these books are not separated, you can count them as one book, so there are 6 books in total which can be arranged in

different ways.

But red and blue books can be rearranged in two ways: blue, red or red, blue, so the number of ways books can be arranged if the red and blue book are not separated is

Hence, the number of ways books can be arranged if the red and blue book are separated is

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}