Integrated Math 3 / Unit 9: Sequences, Series and Functions Review
9.2 Geometric Sequences
Notes
A geometric sequence multiplies by the same common ratio $r$ each time.
- Explicit rule: $a_n = a_1 \cdot r^{n-1}$.
- Example: $2, 6, 18, \dots$ has $r = 3$ and $a_6 = 2 \cdot 3^5 = 486$.
- Finite geometric series: $\displaystyle S_n = \frac{a_1(1 - r^n)}{1 - r}$ for $r \neq 1$.
- Example: $\displaystyle 3 + 6 + 12 + 24 + 48 + 96 = \frac{3(1 - 2^6)}{1 - 2} = 189$.
Practice
What is the 6th term of $2, 6, 18, \dots$?
$a_6 = 2 \cdot 3^5 = 2 \cdot 243 = 486$.
What is the sum of the first 6 terms of $3, 6, 12, \dots$?
$\displaystyle S_6 = \frac{3(1 - 2^6)}{1 - 2} = \frac{3(-63)}{-1} = 189$.
A geometric sequence has positive terms with $a_3 = 12$ and $a_5 = 48$. What is $a_1$?
$\displaystyle \frac{a_5}{a_3} = r^2 = 4$, so $r = 2$ for positive terms. Then $\displaystyle a_1 = \frac{12}{4} = 3$.
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