Because of friction and air resistance, each swing of a pendulum is a little shorter than the previous one. The lengths of the swings form a geometric sequence. Suppose the first swing of the pendulum has an arc length of 100cm and a return swing of 99cm.

On which swing will the length first have a length less than 50 cm?

Find the total distance traveled by the pendulum until it comes to rest.

Since the nth swing has length

100*0.99^(n-1), you want to find n when

100 * 0.99^(n-1) < 50
.99^(n-1) < .5
(n-1) log.99 < log.5
n > 69.9

So, on the 70th swing the arc is less than 50 cm.

Now you know the parameters for the geometric sequence. Just use the sum formula to add up 70 terms

only monney now

Faltu

To determine on which swing the length first becomes less than 50 cm, we can use the given information that the lengths of the swings form a geometric sequence.

Let's denote the length of the first swing as a, and each subsequent swing as a*r, where r is the common ratio.

Given:
First swing length (a) = 100 cm
Return swing length (a*r) = 99 cm

To find the common ratio (r), divide the return swing length by the first swing length:
r = (a*r) / a
r = 99 / 100
r = 0.99

Now, let's determine on which swing the length becomes less than 50 cm.

We can set up an equation:
a * r^n < 50

Plugging in the values we know, we have:
100 * (0.99)^n < 50

To find the value of n, we can solve this equation:

(0.99)^n < 0.5
Taking the logarithm base 0.99 of both sides:
n < log(0.5) / log(0.99)

Using a calculator, we find:
n < 68.97 (approximately)

Therefore, the length of the pendulum will first be less than 50 cm on the 69th swing.

Now, let's find the total distance traveled by the pendulum until it comes to rest.

The distance traveled by the pendulum for each swing is the sum of the arc length and the return swing length.

Since the lengths form a geometric sequence, the sum of the lengths from the first swing until the nth swing can be calculated using the formula:

Sum = a * (1 - r^n) / (1 - r), where a is the first term, r is the common ratio, and n is the number of terms.

Plugging in the values we know:
Sum = 100 * (1 - (0.99)^69) / (1 - 0.99)

Using a calculator, we find:
Sum = 1999.45 cm (approximately)

Therefore, the total distance traveled by the pendulum until it comes to rest is approximately 1999.45 cm.