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Geometric Sequences

CSCA Geometric Sequences study guide organized around the publicly available CSCA syllabus. Practice Mathematics questions on aicsca.com.

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Geometric Sequences

1. Core Concepts

**Definition**: A sequence is called a **Geometric Sequence** if the **ratio** of any term (from the 2nd term) to its preceding term is a constant. This constant is called the **Common Ratio ($q$)**, ($q \neq 0$).

**Mathematical Expression**:

$$\frac{a_{n}}{a_{n-1}} = q \quad (n \ge 2)$$

**Functional Perspective**:

The general term $a_n = a_1 \cdot q^{n-1}$ represents discrete points on an exponential function.

* If $q > 1$: Exponential Growth (e.g., $2, 4, 8...$).

* If $0 < q < 1$: Exponential Decay (e.g., $1, 0.5, 0.25...$).

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2. Key Formulas

#### (1) General Term Formula

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

* **Generalized Form**: $a_n = a_m \cdot q^{n-m}$ (Find term $n$ using term $m$).

#### (2) Summation Formulas ($S_n$)

Let $S_n = a_1 + a_2 + \dots + a_n$.

* **If $q = 1$** (Constant sequence):

$$S_n = n a_1$$

* **If $q \neq 1$**:

$$S_n = \frac{a_1(1-q^n)}{1-q} = \frac{a_1(q^n-1)}{q-1}$$

**Derivation Logic (Multiply and Subtract)**:

This is the standard method: Write $S_n$, write $qS_n$, then subtract the two equations. Most intermediate terms cancel out.

#### (3) Infinite Geometric Series Sum

When $|q| < 1$, as $n \to \infty$, the terms approach 0, and the sum approaches a limit:

$$S_{\infty} = \frac{a_1}{1-q}$$

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3. Key Properties

1. **Geometric Mean**: If $a, b, c$ form a geometric sequence, then $b^2 = ac$.

2. **Index Product Property**: If $m + n = p + k$, then $a_m \cdot a_n = a_p \cdot a_k$.

* E.g., $a_1 \cdot a_6 = a_2 \cdot a_5$.

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4. Typical Examples

**Example 1: Find General Term**

Given $a_2 = 6, a_5 = 48$, find $a_8$.

**Solution**:

Use generalized form: $a_5 = a_2 \cdot q^{5-2}$.

$48 = 6 \cdot q^3 \Rightarrow q^3 = 8 \Rightarrow q = 2$.

Then $a_8 = a_5 \cdot q^{8-5} = 48 \times 2^3 = 384$.

**Example 2: Infinite Sum**

Find the sum: $S = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$.

**Solution**:

$a_1 = 1/2, q = 1/2$.

Since $|q| < 1$, use the infinite sum formula:

$S = \frac{a_1}{1-q} = \frac{1/2}{1 - 1/2} = 1$.

**Example 3: Using Properties**

Given geometric sequence $\{a_n\}$ where $a_1 a_{11} = 20$, find $a_4 a_8$.

**Solution**:

Since indices $1+11 = 12$ and $4+8=12$,

$a_4 a_8 = a_1 a_{11} = 20$.

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5. Common Pitfalls

* **Forgetting $q=1$**: Always check if $q=1$ is possible before using the fraction formula.

* **Sign of q**: If terms alternate signs, $q$ is negative.

* **Solving for q**: When $q^2 = 4$, remember $q = \pm 2$ unless terms are stated to be positive.