PIZZA CRUST Pepe’s Pizza makes 52 pizza crusts the fi rst week and 180 pizza crusts each subsequent week. Dianne’s Delicatessen makes 26 pizza crusts the fi rst week and 90 pizza crusts each subsequent week. In how many weeks will the total number of pizza crusts made by Pepe’s Pizza equal twice the total number of pizza crusts made by Dianne’s Delicatessen?

Always.

Step-by-step explanation:

Let's write two equations.

Pepe's Pizza:

They made 52 pizza crusts the first week and 180 crusts each subsequent week. In other words, in x weeks after the first week, their total number of crusts made is modeled by the equation:

Dianne's Delicatessen:

They made 26 pizza crusts the first week and 90 crusts each subsequent week. So, in x weeks after the first week, their total number of crusts is modeled by:

And we want to find the number of weeks such that the number of crusts made by Pepe's Pizza is equivalent to twice the number of crusts made by Dianne's Delicatessen. In other words:

Substitute P and D for their equations:

Distribute the right:

As we can see, the two equations turns out to be the same.

In other words, no matter what week it is (excluding the first week), the amount of crusts made by Pepe's Pizza will always be twice the crusts made by Dianne's Delicatessen.

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}

i need the options

step-by-step explanation: