Geometric Series and Monkey Swings

The video tutorial explains a problem involving a monkey swinging from a tree. Initially, the monkey swings through an arc of 24 meters, with each subsequent swing covering half the distance of the previous one. The tutorial demonstrates how to express the total distance swung as a geometric series and how to evaluate it using a formula for finite geometric series. Finally, it calculates the total distance the monkey travels after 25 swings, rounding the result to the nearest meter.

What is the initial distance the monkey swings on her first swing?

• 12 meters

• 24 meters

• 6 meters

• 48 meters

How does the distance of each subsequent swing compare to the previous one?

• It triples

• It doubles

• It remains the same

• It is halved

What is the common ratio in the geometric series representing the monkey's swings?

• 1/4

• 1

• 1/3

• 1/2

How is the total distance of the first n swings expressed in terms of a geometric series?

• 24n

• 24(1/2)^n

• 24(1 - (1/2)^n)

• 24(1/2)^(n-1)

What is the formula for the sum of a finite geometric series?

• a(1 - r^n)/(1 - r)

• a(1 + r^n)/(1 + r)

• a(1 + r^n)/(1 - r)

• a(1 - r^n)/(1 + r)

What is the simplified form of the series for the monkey's swings?

• 48(1 + (1/2)^n)

• 24(1 - (1/2)^n)

• 24(1 + (1/2)^n)

• 48(1 - (1/2)^n)

What is the value of 1 minus the common ratio in the simplified formula?

• 2

• 1/4

• 1/2

• 1

What is the total distance the monkey swings after 25 swings?

• 96 meters

• 24 meters

• 48 meters

• 72 meters

Why is the total distance after 25 swings close to 48 meters?

• The series diverges

• The series converges quickly

• The initial distance is small

• The common ratio is large

How does the formula change when calculating the total distance for a specific number of swings?

• The common ratio is squared

• The denominator is halved

• The first term is doubled

• The exponent is adjusted