A pendulum travels a distance of three feet on its first swing. On each successive swing, it travels three-quarters the distance of the previous swing. What is the total distance the pendulum has traveled when it stops swinging?
Let’s think about what we’ve been given here. We know that the very first swing of the pendulum covers a distance of three feet. We can find the distance of the next swing by multiplying three by three-quarters or finding three-quarters of three, which is nine over four or nine-quarters. So on the second swing, it travels a distance of nine-quarters. We repeat this process to find the distance of the third swing. It’s nine-quarters times three-quarters, which is twenty-seven sixteenth.
Now, in fact, this process will continue. And we’re generating a sequence. Each term in our sequence is found by multiplying the previous term by three-quarters. So in fact, this is a geometric sequence. Remember, a geometric sequence is one way each term is found by multiplying the previous one by a fixed nonzero number called the common ratio. Well, in this question, it’s quite clear to us that the common ratio must be three-quarters. And let’s define 𝑎 as our first term. So 𝑎 is equal to three.
Now, we want to find the total distance the pendulum has traveled when it stops swinging. And we do have a little bit of a problem here. If we continue to multiply each term by three-quarters, we never actually get to zero. Remember, the pendulum will have stopped swinging when the term is equal to zero. So instead, we use this formula. It’s the sum to ∞. Now, what happens when the common ratio is between negative one and one? The terms in the sequence approach zero. And so we find the sum to ∞, which is essentially the sum of all the terms in our sequence. It gives us a fixed value. So using the sum to ∞ formula will tell us the total distance the pendulum has traveled when it stops swinging.
Since 𝑎 is equal to three and the common ratio is three-quarters, the sum to ∞ of our sequence is three over one minus three-quarters. One minus three-quarters is one-quarter. So the sum to ∞ is three divided by one-quarter. But of course, dividing by one-quarter is the same as timesing by four. And so we find the sum to ∞ to be equal to 12. And so we say that the total distance that the pendulum travels is 12 feet.