Section 8.2: Geometric Sequences
This section covers geometric sequences, their formulas, and methods to find terms and sums of sequences.
Example 1: Finding a Term in a Geometric Sequence
Given the sequence 2, 6, 18, … find the 6th term.
Step 1: Determine common ratio: \( r = 6 / 2 = 3 \)
Step 2: Use formula \( a_n = a_1 \cdot r^{n-1} \)
Step 3: \( a_6 = 2 \cdot 3^{6-1} = 2 \cdot 243 = 486 \)
Example 2: Sum of a Geometric Sequence
Find the sum of the first 5 terms of 3, 6, 12, …
Step 1: First term \( a_1 = 3 \), common ratio \( r = 2 \)
Step 2: Sum formula for \( r \neq 1 \): \( S_n = a_1 \frac{r^n - 1}{r-1} \)
Step 3: \( S_5 = 3 \frac{2^5 - 1}{2-1} = 3 \cdot 31 = 93 \)
Practice Problems
- Find the 8th term of the sequence 5, 10, 20, …
- Sum the first 6 terms of 1, 3, 9, …
- Given \( a_1 = 4 \) and \( r = 2 \), find \( a_{10} \)
- Find the sum of the first 7 terms of 2, 6, 18, …
- Determine the 5th term if the 2nd term is 12 and the 4th term is 108