Ethan needs to save at least $500 to purchase a new dirt bike. so far, he had saved $175. if he hopes to use two-fifths of his next paycheck to cover the amount, how much money must he make in his paycheck?

he must make $812.5 or more in his paycheck

Step-by-step explanation:

Let x represent the amount if money that Ethan must make on his next paycheck.

Ethan needs to save at least $500 to purchase a new dirt bike. So far, he had saved $175. If he hopes to use two-fifths of his next paycheck to cover the amount, then the equation becomes

175 + 2x/5 ≥ 500

2x/5 ≥ 500 - 175

2x/5 ≥ 325

2x ≥ 325 × 5

2x ≥ 1625

x ≥ 1625/2

x ≥ 812.5

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}