We are told that a pendulum initially swings through an arc that is 20 inches long and the length of the arc decreases by a factor of 0.85 on each swing.
We want to write a recursive rule to represent the sequence of the lengths of the arc. To do so, we will recall the general recursive formula for a geometric sequence.
a_n = a_(n-1) * r for n > 1
In this form r is the common ratio of the sequence. Since the length of the arc is 0.85 of previous swing, our common ratio is 0.85. Therefore, we will substitute r= 0.85 into the recursive formula.
a_n = a_(n-1) * 0.85 for n > 1
Great! Since we are given the initial length of the arc as p_1= 20, let's rearrange our formula by switching a with p.
a_n = a_(n-1) * 0.85 for n > 1
⇕
p_n = p_(n-1) * 0.85 for n > 1
b
This time we will use the explicit formula to represent the sequence of the lengths of the arc. Let's begin by recalling the general explicit rule for a geometric sequence.
a_n = a_1 * r^(n-1) for n ≥ 1
In this form a_1 represents the first term and r is the common ratio of the sequence. Since we are given that p_1= 20 and r= 0.85, we can write our explicit formula by switching a with p. Let's substitute our values!
p_n = 20 * ( 0.85)^(n-1) for n ≥ 1
c
Now we will find the approximate total distance the pendulum has swung after 11 swings. To do so, let's first remember our sequence.
We want to find the sum of the terms of the above geometric sequence. Therefore, we need to use the formula for the sum of a finite geometric series.
S_n = a_1(1- r^n)/1- r
In this formula n represents the number of terms, a_1 represents the first term, and r represents the common ratio. Since we want to find the total distance after 11 swings, let's substitute n= 11, a_1= 20, and r= 0.85 into the formula and calculate it using a calculator.
The approximate total distance after 11 swings is 111 inches.
d
Last, we will find the approximate total distance that the pendulum has swung when it stops. Notice that our common ratio is between 0 and 1.
0 < 0.85 < 1Therefore, the lengths of the arc will diminish rapidly while it is approaching the stop. In other words, although the situation represents an infinite geometric series, finally we will have a finite sum. Let's recall the summation formula of an infinite geometric series for | r|>0.
S_n = a_1/1- r
Now, we can substitute a_1= 20 and r= 0.85 into the formula and calculate the total distance.
