Geometric Sequences

Patterns with common ratios

Geometric Sequences

Definition

A geometric sequence has a constant ratio between consecutive terms.

Example: $3, 6, 12, 24, 48, ...$

Common ratio: $r = 2$

Finding the Common Ratio

$r = \frac{a_2}{a_1} = \frac{a_3}{a_2} = ...$

Divide any term by the previous term.

Explicit Formula

To find the $n$ th term: $a_n = a_1 \cdot r^{n-1}$

where:

$a_n$ = nth term
$a_1$ = first term
$r$ = common ratio
$n$ = term number

Recursive Formula

$a_n = a_{n-1} \cdot r$

Each term equals the previous term times $r$ .

Growth vs. Decay

Growth: $|r| > 1$

Terms get larger in magnitude

Decay: $0 < |r| < 1$

Terms get smaller

Real-World Applications

Compound interest
Population growth
Radioactive decay

📚 Practice Problems

1Problem 1easy

❓ Question:

Find the common ratio: $5, 15, 45, 135, ...$

💡 Show Solution

Divide consecutive terms:

$r = \frac{15}{5} = 3$

Check: $\frac{45}{15} = 3$ ✓

Answer: Common ratio = $3$

2Problem 2medium

❓ Question:

Find the 8th term of the sequence: $2, 6, 18, 54, ...$

💡 Show Solution

Step 1: Identify $a_1$ and $r$ $a_1 = 2, \quad r = \frac{6}{2} = 3$

Step 2: Use the explicit formula $a_n = a_1 \cdot r^{n-1}$

Step 3: Substitute $n = 8$ $a_8 = 2 \cdot 3^{8-1}$ $= 2 \cdot 3^7$ $= 2 \cdot 2187$ $= 4374$

Answer: $a_8 = 4374$

3Problem 3hard

❓ Question:

The 3rd term of a geometric sequence is 12 and the 6th term is 96. Find the first term and common ratio.

💡 Show Solution

Given: $a_3 = 12$ and $a_6 = 96$

Step 1: Write equations using $a_n = a_1 \cdot r^{n-1}$ $a_3: \quad 12 = a_1 \cdot r^2$ $a_6: \quad 96 = a_1 \cdot r^5$

Step 2: Divide the second by the first $\frac{96}{12} = \frac{a_1 \cdot r^5}{a_1 \cdot r^2}$ $8 = r^3$ $r = 2$

Step 3: Find $a_1$ using $a_3 = 12$ $12 = a_1 \cdot 2^2$ $12 = 4a_1$ $a_1 = 3$

Check: Sequence is $3, 6, 12, 24, 48, 96, ...$ ✓

Answer: $a_1 = 3$ , $r = 2$

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