College Algebra & Trigonometry, 2018a

300 CHAPTER 6. SEQUENCES AND SERIES
Geometric Series:
Given a geometric series, whose first term is a and with a constant ratio of r,
n∑
a ∗ r k−1 , we can write out the terms of the series in a similar way that we did
k=1
for the arithmetic series.
n∑
a ∗ r k−1 = a + ar + ar 2 + ar 3 + ···+ ar n−1
k=1
The trick to finding a formula for the sum of this type of series is to multiply both
sides of the previous equation by r.
For simplicity’s sake let’s rename the sum of the series
So,
S n = a + ar + ar 2 + ar 3 + ···+ ar n−1
and
n∑
a ∗ r k−1 as S n .
r ∗ S n = r(a + ar + ar 2 + ar 3 + ···+ ar n−1 )=ar + ar 2 + ar 3 + ···+ ar n
If we subtract these two equations, we will have:
S n = a + ar + ar 2 + ar 3 + ···+ ar n−1
− ( r ∗ S n = ar + ar 2 + ar 3 + ···+ ar n)
Then we’ll have:
S n − rS n = a − ar n
k=1