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8.2. GEOMETRIC SERIES
In this activity we will find a shortcut formula for S n that does not involve a sum of n
terms.
(a) Multiply S n by r. What does the resulting sum look like?
(b) Subtract r S n from S n and explain why
S n − r S n = a − ar
n .
(8.2)
(c) Solve equation (8.2) for S n to find a simple formula for S n that does not involve
adding n terms.
⊳
We can summarize the result of Activity 8.4 in the following way.
A geometric sum S n is a sum of the form
S n = a + ar + ar
2 + · · · + ar
n−1
,
(8.3)
where a and r are real numbers such that r 1. The geometric sum S n can be written
more simply as
S n = a + ar + ar
2 + · · · + ar
n−1 =
a(1 − r n )
1 − r
.
(8.4)
We now apply equation (8.4) to the example involving warfarin from Preview Activity 8.2. Recall that
Q(n) = (5 × 0.08)
1 + 0.08 + 0.08
2 + 0.08
3 + · · · + 0.08
n−1 mg,
so Q(n) is a geometric sum with a = 5 × 0.08 = 0.4 and r = 0.08. Thus,
Q(n) = 0.4
1 − 0.08 n
1 − 0.08
=
1
2.3
(1 − 0.08
n ) .
Notice that as n goes to infinity, the value of 0.08 n goes to 0. So,
lim
n→∞
Q(n) = lim
n→∞
1
2.3
(1 − 0.08
n ) =
1
2.3
≈ 0.435.
Therefore, the long-term level of Warfarin in the blood under these conditions is
1
2.3 , which
is approximately 0.435 mg.
To determine the long-term effect of Warfarin, we considered a geometric sum of n
terms, and then considered what happened as n was allowed to grow without bound. In
this sense, we were actually interested in an infinite geometric sum (the result of letting n
go to infinity in the finite sum). We call such an infinite geometric sum a geometric series.