Test Your Knowledge: Understanding & Writing Recursive Formulas

📚 Quick Study Guide

• 🔢 Definition: A recursive formula defines a sequence where each term is defined based on the preceding terms.
• ➕ Initial Term: You must have at least one initial term defined (e.g., $a_1$ or $a_0$).
• 🔄 Recursive Step: This is the formula that relates a term $a_n$ to one or more previous terms (e.g., $a_{n-1}$, $a_{n-2}$).
• 📝 General Form: A common structure looks like this: $a_n = f(a_{n-1})$, where $f$ is some function.
• 💡 Example 1: Arithmetic sequence: $a_n = a_{n-1} + d$, where $d$ is the common difference.
• ➗ Example 2: Geometric sequence: $a_n = r * a_{n-1}$, where $r$ is the common ratio.
• 🌱 Writing Recursive Formulas: Identify the pattern between consecutive terms. Express the current term based on the previous term(s). Clearly define the initial term(s).

Practice Quiz

1. What is the next term in the sequence defined by $a_1 = 3$ and $a_n = 2a_{n-1} - 1$?
1. 1
2. 3
3. 5
4. 7
2. Which of the following is a recursive formula for the sequence 2, 4, 8, 16, ...?
1. $a_n = a_{n-1} + 2$, $a_1 = 2$
2. $a_n = 2a_{n-1}$, $a_1 = 2$
3. $a_n = a_{n-1}^2$, $a_1 = 2$
4. $a_n = 2 + a_{n-1}$, $a_1 = 2$
3. Given the recursive formula $a_n = a_{n-1} + a_{n-2}$ with $a_1 = 1$ and $a_2 = 1$, what is $a_5$?
1. 3
2. 5
3. 8
4. 13
4. What is the initial term needed to fully define the sequence given the recursive formula $a_n = 5a_{n-1} + 2$?
1. $a_0$
2. $a_1$
3. $a_2$
4. $a_n$
5. Which recursive formula represents the sequence of odd numbers starting with 1 (1, 3, 5, 7,...)?
1. $a_n = a_{n-1} + 1$, $a_1 = 1$
2. $a_n = a_{n-1} + 2$, $a_1 = 1$
3. $a_n = 2a_{n-1} + 1$, $a_1 = 1$
4. $a_n = a_{n-1} - 2$, $a_1 = 1$
6. If $a_n = n + a_{n-1}$ and $a_0 = 1$, find $a_3$.
1. 3
2. 6
3. 7
4. 10
7. What does $a_n = a_{n-1} * n$ represent with $a_1 = 1$?
1. The sum of all previous terms
2. The nth root of the previous term
3. The factorial of n
4. The nth power of the previous term