Question

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 100 cm and a return swing of 99 cm.
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.

Answer 1

Im just stuck on this one problem.

Answer 2

sounds like 100 is the first term of the geometric sequence and 99 is the second term

Answer 3

if so, then

common ratio = (second term)/(first term)

r = (second term)/(first term)

r = 99/100

r = 0.99

Answer 4

Your nth term would then be

an = a*r^(n-1)

an = 100*0.99^(n-1)

Answer 5

Now plug in an = 50 and solve for n to get your answer

Answer 6

plug in 50? an= 100*0.99^50-1?

Answer 7

no into an, not n

Answer 8

an = 100*0.99^(n-1)

50 = 100*0.99^(n-1)

now solve for n

Answer 9

99^n-1?

Answer 10

:( dang..

Answer 11

50 = 100*0.99^(n-1)

50/100 = 0.99^(n-1)

0.5 = 0.99^(n-1)

ln(0.5) = ln(0.99^(n-1))

ln(0.5) = (n-1)ln(0.99)

ln(0.5) = n*ln(0.99)-ln(0.99)

ln(0.5)+ln(0.99) = n*ln(0.99)

( ln(0.5)+ln(0.99) )/ln(0.99) = n

n = ( ln(0.5)+ln(0.99) )/ln(0.99)

n = 69.9675639365284

n = 70 (round up)

So on the 70th swing (and up), the arc length will be less than 50