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### Course: Get ready for Precalculus>Unit 6

Lesson 6: Geometric series

# Geometric series word problems: swing

The length a monkey passes when swinging from a tree can be modeled by a geometric series!  We learn that each swing's length is half the previous one, forming a finite geometric series. By applying the sum formula, we calculate the total distance the monkey swings.

## Want to join the conversation?

• if you swing a pendulum, and each time it swings 1/2 of the distance, would it go on forever?
• Yes - theoretically, it would never reach zero. It would technically just approach zero.
• That is a cockroach Sal not a monkey :D
• frfrfrfrfr
• Physically, isn't 0.5 of the distance of the previous swing impossible ?
• I think so too. 1/2 the distance of the previous swing means the pendulum would come to a stop exactly at the bottom of the swing. You'd only get that if you move a wall at that position to crash into after the first swing, or if you have some external force acting on the pendulum.
• please help me with this question:- A swinging pendulum covers 32 centimetres in the first swing, 24 centimetres in the second swing and 18 centimetres in the third and so on. what is the total distance it swings before coming to rest?
• You can create a geometric series based on this. a1 is 32 centimeters, r = 0.75. Now you need to sum the series up. I believe that you need to treat this as an infinite geometric series and use the formula for an infinite geometric series. a1/(1-r). So plug in the values and see what the answer is. This is my understanding and I hope this helps you!
• What if we were given the final distance the monkey traveled but we weren't given the nth term. How would we solve for n?
• you could possibly use the sum formula and start from there
• please help! the first question asks the total length of the swing after the nth swing right? so that should be a very small length as the swing length continuously decreases.

and in the second question, the distance after her 25h swing, the answer is 48m, not a really small number. so...

in that case, we should not use that formula to answer the first question? or the first question should be something like the total length of all the swings added?
• Total length (or total distance) is already saying total length of all swings added. if it just wanted the length of the 25th swing it would ask something like "what is the length of the 25th swing". Saying total length is saying it wants all swings added up.
• Can anyone help me understand why at the last step, which (forgive me, I can't type fractions here) is 48(1 - 1/2 ^25), you can't first distribute the 48 to both terms inside the parentheses and get an equivalent answer to the one you get when you evaluate it as is? I know we do computation inside parentheses first, I'm just wondering why that doesn't work. Or am I just entering it into my calculator wrong? Or does it have something to do with that fraction raised to a power?
• We can distribute the 48, but because of the exponent it's hardly worth the trouble.

48(1 − (1∕2)^25)

= 48 − 48 ∙ (1∕2)^25

= 48 − 3 ∙ 2^4 ∙ (1∕2)^4 ∙ (1∕2)^21

= 48 − 3 ∙ 2^4 ∙ 2^(−4) ∙ (1∕2)^21

= 48 − 3 ∙ (1∕2)^21
• in a-ar^2, why are we subracting "what would be the first term after n-1?
• Am I the only one that's been doing this for 3 hours and still doesn't understand
(1 vote)
• In this problem, a monkey initially swings for 24m. Then on the next pass, it swings half the first length, which means first swing length * 0.5, i.e. 12m. After that it continues to decrease by half.
So on the third pass, we get
12 * 0.5 = 6m
On the fourth pass, we get
6 * 0.5 = 3m
And so on.

Now notice on the second pass, we get
24 * 0.5
On the third pass, we get
(24 * 0.5) * 0.5
= 24 * 0.5²
On the fourth pass, we get
(24 * 0.5 * 0.5) * 0.5
= 24 * 0.5³
The power keep increasing.

So we can claim, on the n-th pass, the power is (n - 1), and the length on the n-th pass is 24 * 0.5ⁿ⁻¹.

A few lessons ago, we've learnt the Finite Geometric series formula
``S = [a * (rⁿ - 1)] / (r - 1)``

where a is the first term, r is the common ratio and n is the number of terms.

Since we are required to represent S in terms of n, we now only have to find a and r.
a = 24m, which is provided in the problem statement.
r = 0.5, which we observed previously.
Therefore,
``S = 24 * (0.5ⁿ - 1) / (0.5 - 1)S = 48 * (1 - 0.5ⁿ)``

To find out the total distance the monkey travelled on the 25th swing, simply substitute n by 25 and you get your answer.