Initially a pendulum swings through an arc of 2 feet On each

Initially, a pendulum swings through an arc of 2 feet. On each successive swing, the length of the arc is 0.9 of the previous arc. a) What is the length of the arc of the 10^th swing? b) After 15 swings, what total length will the pendulum have swung? c) When it stops, what total length will the pendulum have swung?

Solution

a)The sequence is

2, 2 *(0.9), 2*(0.9)², 2*(0.9)³, .....

U_n = 2*(0.9)^(n - 1)

So U_10 = 2*(0.9)^9

=0.7748 feet

b) After 15 swings, what total length will the pendulum have swung?

S_n = U_1 + U_2 + U_3 + .... + U_n

= a(1 - r^n)/(1 - r) Where a = U_1, r = 0.9, n = number of terms being summed)

So S_15 = 2(1 - (0.9)^15)/(1 - 0.9)
= 15.8822 ft

c)In the limit (0.9)^n 0

So the limiting sum = a/(1 - r)
= 2/(1 - 0.9)
= 2/(0.1)
= 20\'

ie when it stops it will have travelled 20 ft