More than Just Lines on a Map: Best Practices for U.S Bike Routes
05-LS-Sum of a Geometric Sequence.docx
1. Sum of a Geometric Sequence
For a geometric sequence 𝑆𝑛 =
𝑎(𝑟𝑛−1)
𝑟−1
proof:
we know that: 𝑆𝑛 = 𝑎 + 𝑎𝑟 + 𝑎𝑟2
+ ⋯ + 𝑎𝑟𝑛−1
if we multiply both sides by r we have: 𝑟𝑆𝑛 = 𝑎𝑟 + 𝑎𝑟2
+ 𝑎𝑟3
+ ⋯ + 𝑎𝑟𝑛
find an expression for rSn-Sn: 𝑟𝑆𝑛 − 𝑆𝑛 = 𝑎𝑟𝑛
− 𝑎
now common factor out Sn: 𝑆𝑛 (𝑟 − 1) = 𝑎𝑟𝑛
− 𝑎
now divide by r-1 on both sides: 𝑆𝑛 =
𝑎𝑟𝑛−𝑎
𝑟−1
common fact out the ‘a’ in the numerator: 𝑆𝑛 =
𝑎(𝑟𝑛−1)
𝑟−1
QED
Ex. Find S8 for the given sequence.
a) 5+15+45+... b) 50, 25,
25
2
, …
Ex. Evaluate: 6+18+...+4374