Integrated Math 1 / Unit 5: Exponential Functions and Geometric Sequences
5.4 Geometric Sequences
Notes
A geometric sequence multiplies by the same number, the common ratio $r$, each time.
- Find $r$ by dividing a term by the previous one.
- Explicit formula: $a_n = a_1 \cdot r^{n-1}$. Recursive: $a_n = r \cdot a_{n-1}$.
- Example: $3, 6, 12, 24, \dots$ has $r = 2$ and $a_5 = 3 \cdot 2^4 = 48$.
- A geometric sequence is an exponential function whose inputs are whole numbers.
Practice
What is the common ratio of $81, 27, 9, 3, \dots$?
$\displaystyle 27 \div 81 = \frac{1}{3}$.
A sequence is given by $a_n = 5 \cdot 3^{n-1}$. What is $a_4$?
$5 \cdot 3^3 = 5 \cdot 27 = 135$.
Which sequence is geometric?
Each term in $2, 6, 18, 54$ is 3 times the one before.
Ready for more? Take the Unit 5 test.