AP Precalculus Practice Test: Unit 2 Question #3 Rewriting the General Term of a Sequence

My AP Precalculus Practice Tests are carefully designed to help students build confidence for in-class assessments, support their work on AP Classroom assignments, and thoroughly prepare them for the AP Precalculus exam in May.

### AP Precalculus Practice Test: Unit 2, Question #3
**Topic:** Rewriting the General Term of a Sequence

This question focuses on transforming or rewriting the general term of a sequence (arithmetic or geometric) into a simplified or alternative form. The goal is to ensure the term clearly represents the sequence’s structure and relationships.

---

**Example Problem:**
The general term of a sequence is given as:
\[
a_n = 3 + 4(n-1)
\]
Rewrite the expression to simplify and explicitly show the linear relationship.

---

**Steps to Solve:**

1. **Expand the equation to simplify the expression:**
\[
a_n = 3 + 4(n-1)
\]
Distribute \(4\) to \((n-1)\):
\[
a_n = 3 + 4n - 4
\]

2. **Combine like terms:**
\[
a_n = 4n - 1
\]

3. **Rewrite the general term in its simplified form:**
\[
a_n = 4n - 1
\]

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**Correct Answer:**
\[
a_n = 4n - 1
\]

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### General Guidelines for Rewriting Sequence Terms

#### **Arithmetic Sequence**
- Start from the formula \( a_n = a_1 + (n-1)d \).
- Expand and simplify as needed to express \( a_n \) in terms of \( n \).

#### **Geometric Sequence**
- Use the formula \( g_n = g_1 \cdot r^{n-1} \).
- Simplify exponential expressions, if possible, to a compact form.

---

**Example for Geometric Sequences:**
Given \( g_n = 2 \cdot 3^{n-1} \), rewrite it in terms of its growth pattern:
\[
g_n = 2 \cdot 3^{n-1}
\]
No further simplification is typically required unless asked to evaluate specific terms or combine constants.

---

**Potential Multiple-Choice Options (Arithmetic Example):**
A. \( a_n = 4n - 1 \)
B. \( a_n = 3 + 4n \)
C. \( a_n = 4n + 3 \)
D. \( a_n = 4n - 4 \)

(Answer: **A**)

**Tips:**
- Ensure any rewritten form maintains equivalence to the original formula.
- For arithmetic sequences, clearly separate terms involving \(n\) from constants.
- For geometric sequences, maintain clarity in exponential expressions.

This problem emphasizes clarity and understanding of sequence structure, preparing students for analyzing patterns in data and formulas.

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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