Jeff uses 116rubber bands to make a rubber band ball. he estimates that he has used 448 rubber bands to make 12 rubber band bottles of the same size. use estimation to check and explain whether his answer is reasonable or not.

His answer is not reasonable.

Step-by-step explanation:

448 rubber bands to make 12 rubber band bottles of the same size.

450 divided by 10  is about 45 rubber band per rubber band bottle.

This is nowhere near the the 116 rubber bands to make a rubber band ball.

If the are about the same size, we should use about the same number of rubber bands.

If he used about 100 rubber bands per rubber band bottle * 12 bottles, that would be about 1200 rubber bands.

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}