Geometric series

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Vikash

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A ball is dropped from height of 10 m. It bounces to a height of 7 m
and continues in geometric sequence. How high will it bounce after
the 4th bounce?

I know the answer must be:

10 x 0.7 = 7 (first bounce)
7 x 0.7 = 4.9 (second bounce)
4.9 x 0.7 = 3.43 (third bounce)
3.43 x 0.7 = 2.401 (fourth bounce)

but if I use formula ar^(n-1) - this will be 10 x (0.7^3) for the
forth bounce and that is incorrect.

So my real (general) question is: How do I know when to use n vs n-1
--or why in this question do I use n (i.e. 4) and not n-1 (i.e 3)?

I can figure out these questions by NOT using the formula but just
going through the sequences. But I need to figure out the general logic.
 
GREAT question.

The height of the first bounce is as you say 7. So, for the height of the fourth bounce, n = 4. a is 7, and r is 0.7

[MATH]7 * 0.7^{(4-1)} = 7 * 0.7^3 = 7 * 0.343 = 2.401[/MATH]
Or we could analyze heights.

The initial height is 10. The height after the fourth bounce is the fifth height, and n = 5, a is 10, and r is still 0.7.

[MATH]10 * 0.7^{(5-1)} = 10 * 3^4 = 10 * 0.2401 = 2.401[/MATH]
The formula will work just fine once you become certain what it is you are counting.
 
Last edited:
Vikash said:
A ball is dropped from height of 10 m. It bounces to a height of 7 m
and continues in geometric sequence. How high will it bounce after
the 4th bounce?

I know the answer must be:

10 x 0.7 = 7 (first bounce)
7 x 0.7 = 4.9 (second bounce)
4.9 x 0.7 = 3.43 (third bounce)
3.43 x 0.7 = 2.401 (fourth bounce)

but if I use formula ar^(n-1) - this will be 10 x (0.7^3) for the
forth bounce and that is incorrect.

So my real (general) question is: How do I know when to use n vs n-1
--or why in this question do I use n (i.e. 4) and not n-1 (i.e 3)?

I can figure out these questions by NOT using the formula but just
going through the sequences. But I need to figure out the general logic.
Click to expand...
If you use ar^(n-1) with a = 10, then n is not the number of the bounce, as the first bounce is not 10 but 7, just as you said.

Taking a=10, you are making a sequence starting with n=0, for which the proper formula is ar^n.

Off-by-one errors commonly occur this way. You need to identify whether you are starting a count at 0 or 1, and make the formula appropriately. I commonly do this by checking the formula I am using for the first term, and making sure it gives the correct result.

When you apply a formula without thinking, you get in trouble. Formulas plus thinking work!
 
Dr.Peterson said:
If you use ar^(n-1) with a = 10, then n is not the number of the bounce, as the first bounce is not 10 but 7, just as you said.

Taking a=10, you are making a sequence starting with n=0, for which the proper formula is ar^n.

Off-by-one errors commonly occur this way. You need to identify whether you are starting a count at 0 or 1, and make the formula appropriately. I commonly do this by checking the formula I am using for the first term, and making sure it gives the correct result.

When you apply a formula without thinking, you get in trouble. Formulas plus thinking work!
Click to expand...
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