On a particular day, the wind added 5 miles per hour to alfonso's rate when he was cycling with the wind and subtracted 5 miles per hour from his rate on his return trip. alfonso found that in the same amount of time he could cycle 50 miles with the wind, he could go only 30 miles against the wind. what is his normal bicycling speed with no wind?

speed with no wind = s

speed with wind = s + 5

speed against wind = s - 5

time = t

speed = distance/time

distance = speed * time

With wind, time = t, speed = s + 5, distance = 50

50 = (s + 5)t

Against wind, time = t, speed = s - 5, distance = 30

30 = (s - 5)t

t = 30/(s - 5)

50 = (s + 5) * 30/(s - 5)

50(s - 5) = (s + 5) * 30

50s - 250 = 30s + 150

20s = 400

s = 20

20 mph

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}

16

step-by-step explanation: