Precalculus / Unit 10: Sequences, Series and Intro to Limits
10.2 Geometric Sequences and Series
Notes
A geometric sequence multiplies by the same common ratio $r$ each time.
- $a_n = a_1 r^{n - 1}$.
- The sum of the first $n$ terms: $\displaystyle S_n = \frac{a_1(1 - r^n)}{1 - r}$ for $r \ne 1$.
- Example: $3, 6, 12, \dots$ has $a_8 = 3 \cdot 2^7 = 384$.
- Example: $\displaystyle 2 + 6 + 18 + 54 + 162 = \frac{2(1 - 3^5)}{1 - 3} = 242$.
Practice
What is the 6th term of $81, 27, 9, \dots$?
$\displaystyle r = \frac{1}{3}$, so $\displaystyle a_6 = 81 \cdot \left(\frac{1}{3}\right)^5 = \frac{81}{243} = \frac{1}{3}$.
What is the sum of the first 5 terms of $2, 6, 18, \dots$?
$\displaystyle S_5 = \frac{2(1 - 243)}{1 - 3} = 242$.
The 3rd term of a geometric sequence is 12 and the 5th term is 48 (with positive ratio). What is the common ratio?
$\displaystyle r^2 = \frac{48}{12} = 4$, so $r = 2$.
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