Algebra 2 / Unit 7: Sequences and Series
7.2 Geometric Sequences
Notes
A geometric sequence multiplies by the same common ratio $r$ each time.
- Explicit formula: $a_n = a_1 \cdot r^{n - 1}$.
- Find $r$ by dividing any term by the one before it.
- Example: $2, 10, 50, \dots$ has $r = 5$, so $a_n = 2 \cdot 5^{n - 1}$ and $a_5 = 2 \cdot 625 = 1250$.
- A negative $r$ makes the signs alternate.
Practice
Find the 8th term of $3, 6, 12, 24, \dots$
$a_8 = 3 \cdot 2^7 = 3 \cdot 128 = 384$.
What is the common ratio of $81, -27, 9, -3, \dots$?
$\displaystyle \frac{-27}{81} = -\frac{1}{3}$.
A geometric sequence has $a_1 = 5$ and $r = 3$. Which is its explicit formula?
Use $a_n = a_1 r^{n - 1}$.
Ready for more? Take the Unit 7 test.