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Do I Use n Or n-1 to Find the nth Term in a Geometric Sequence?

Date: 12/01/2009 at 03:50:01
From: Adrian
Subject: geometric sequences - when to use n or n-1?

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. 

PS:  You guys are great--I have asked 3 questions in past 2 months
and you come back to me with brilliant explanations.  I will make a
$10 Xmas present to Dr. Math!  Thanks again.

Date: 12/01/2009 at 09:23:54
From: Doctor Peterson
Subject: Re: geometric sequences - when to use n or n-1?

Hi, Adrian.

You have essentially given the answer yourself; you don't just 
blindly apply a formula, but think about what is happening in a 
specific problem.  In your example, each bounce multiplies the height 
by 0.7, so you are multiplying the initial height by 0.7 raised to 
the number of bounces.

Relating this to the formula, the issue is what "n" MEANS.  Here, 
it's the number of bounces--the number of terms AFTER the first, 
initial height.  So the initial height is the height for n = 0 (after 
zero bounces).  The sequence here is

  n:    0     1     2    ...   n
  a_n: ar^0, ar^1, ar^2, ..., ar^n

When the terms are numbered starting at n = 1, you need to use n-1 in 
place of the n above, since n is one more than it was there:

  n:    1     2     3    ...   n
  a_n: ar^0, ar^1, ar^2, ..., ar^(n-1)

Students often try to use formulas as a magic incantation, without 
giving any thought to the meaning of the variables.  They work well 
only when they are applied in line with the definitions and 
conditions that come with them; too often people ignore these.  In 
this case, your formula for a term is part of the following fuller 

  The nth term of a geometric sequence whose first term is a, and
  whose common ratio is r, is a_n = ar^(n-1).

If a is the zeroth term, not the first term, this has to be modified 
accordingly.  If you wanted to apply this formula as is to your 
problem, you would note that the FIRST term, the initial height, is 
the height after zero bounces, and the height after one bounce is 
the second term, and so on; so the height after 4 bounces is the 
(4+1)th term.  Take n = 5 and it will work.

Sometimes the formula is stated with the idea that the first term is 
for n = 0 or n = 1 explicit, using subscripts:

  If a_0 is the first term of a geometric sequence with common ratio
  r, then a_n = a_0 r^n.

  If a_1 is the first term of a geometric sequence with common ratio
  r, then a_n = a_1 r^(n-1).

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Sequences, Series

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