等比数列děngbǐ shùliè

Core Concept

A geometric sequence is a sequence where, starting from the second term, the ratio of each term to its preceding term equals the same constant. This constant is called the common ratio, typically denoted by $q$.

Mathematical Definition

For a sequence $\{a_n\}$, if there exists a constant $q \neq 0$ such that:

$\frac{a_{n+1}}{a_n} = q \quad (n \in \mathbb{N}^*, a_n \neq 0)$

then $\{a_n\}$ is called a geometric sequence with common ratio $q$.

General Term Formula

$a_n = a_1 \cdot q^{n-1}$

where:

• $a_1$ is the first term
• $q$ is the common ratio
• $n$ is the term number

Sum Formula

When $q \neq 1$: $S_n = \frac{a_1(1 - q^n)}{1 - q} = \frac{a_1 - a_n q}{1 - q}$

When $q = 1$: $S_n = n \cdot a_1$

Geometric vs Arithmetic Sequences

Feature	Geometric	Arithmetic
Definition	Ratio of consecutive terms is constant	Difference of consecutive terms is constant
Notation	$\frac{a_{n+1}}{a_n} = q$	$a_{n+1} - a_n = d$
General Term	$a_n = a_1 \cdot q^{n-1}$	$a_n = a_1 + (n-1)d$
Mean	$b^2 = ac$ (geometric mean)	$b = \frac{a+c}{2}$ (arithmetic mean)

Real-World Applications

Application 1: Cell Division

Problem: A cell divides every hour into 2 cells. How many cells after 8 hours?

Solution:

• First term $a_1 = 1$
• Common ratio $q = 2$
• After 8 hours: $a_9 = 1 \times 2^{8} = 256$ cells

Application 2: Compound Interest

Problem: $10,000 deposited at 5% annual interest (compound). Total after 10 years?

Solution: $a_{11} = 10000 \times 1.05^{10} \approx \$16,288.95$

Application 3: Radioactive Decay

Problem: Substance decays 20% annually. Initial mass 100g, remaining after 5 years?

Solution: $a_6 = 100 \times 0.8^5 = 32.768 \text{ g}$

CSCA Practice Problems

💡 Note: The following practice problems are designed based on the CSCA exam syllabus and Chinese standardized test formats to help students familiarize themselves with question types and problem-solving approaches.

Example 1: Basic (Difficulty ★★☆☆☆)

In geometric sequence $\{a_n\}$, $a_2 = 6$ and $a_5 = 48$. Find common ratio $q$.

Options:

• A. 2
• B. 3
• C. 4
• D. 8

Detailed Solution:

$a_5 = a_2 \cdot q^{3}$ $48 = 6 \cdot q^3$ $q^3 = 8$ $q = 2$

Answer: A

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Example 2: Intermediate (Difficulty ★★★☆☆)

In geometric sequence $\{a_n\}$, $a_1 + a_2 = 3$ and $a_2 + a_3 = 6$. Find $a_5$.

Detailed Solution:

$a_1(1 + q) = 3$ ... ① $a_1 q(1 + q) = 6$ ... ②

Divide ②÷①: $q = 2$

Substitute into ①: $a_1 = 1$

Therefore: $a_5 = 1 \times 2^4 = 16$

Answer: 16

Common Mistakes

❌ Mistake 1: Geometric sequences always increase

Correction: Growth depends on both $a_1$ and $q$:

• $a_1 > 0, q > 1$ → increasing
• $a_1 > 0, 0 < q < 1$ → decreasing
• $q < 0$ → alternating signs

❌ Mistake 2: Common ratio can be zero

Correction: $q \neq 0$, otherwise all terms from the second onward would be zero.

❌ Mistake 3: Confusing geometric and arithmetic means

Correction:

• Geometric mean: $b^2 = ac$
• Arithmetic mean: $b = \frac{a+c}{2}$

Don't mix them up!

❌ Mistake 4: Forgetting to classify when summing

Correction: Always consider $q = 1$ and $q \neq 1$ separately when finding sums.

Study Tips

1. ✅ Compare with arithmetic sequences: Understand "ratio" vs "difference"
2. ✅ Master formulas: Memorize general term and sum formulas
3. ✅ Case analysis: Consider different cases for $q$
4. ✅ Real applications: Cell division, compound interest, decay are typical models

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💡 Exam Tip: Geometric and arithmetic sequences are equally important in CSCA exams, each accounting for about 50% of sequence problems. Study them comparatively!