Sequences
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. What is a Sequence?
Definition:
A sequence is an ordered list of numbers following a specific pattern
General Notation:
\[ a_1, a_2, a_3, a_4, \ldots, a_n, \ldots \]
• \( a_n \) represents the nth term (general term)
• \( n \) is the position number (index)
• \( a_1 \) is the first term
2. Explicit Formula
Definition:
An explicit formula directly calculates any term in the sequence using only the position number \( n \)
\[ a_n = f(n) \]
Key Features:
• Finds any term directly without previous terms
• Only depends on \( n \) (position)
• Also called "closed form"
Examples:
Formula: \( a_n = 2n + 1 \)
Find the first 4 terms:
\( a_1 = 2(1) + 1 = 3 \)
\( a_2 = 2(2) + 1 = 5 \)
\( a_3 = 2(3) + 1 = 7 \)
\( a_4 = 2(4) + 1 = 9 \)
Sequence: 3, 5, 7, 9, ...
Formula: \( a_n = n^2 - 1 \)
Find the 5th term:
\( a_5 = 5^2 - 1 = 25 - 1 = 24 \)
Can find any term without calculating previous terms!
3. Recursive Formula
Definition:
A recursive formula defines each term using one or more previous terms
\[ a_n = f(a_{n-1}) \text{ or } a_n = f(a_{n-1}, a_{n-2}, \ldots) \]
Two Parts Required:
1. Initial condition(s): Starting term(s) like \( a_1 = 5 \)
2. Recursive equation: How to get next term from previous
Examples:
Formula: \( a_1 = 3, \quad a_n = a_{n-1} + 2 \)
Find the first 5 terms:
\( a_1 = 3 \) (given)
\( a_2 = a_1 + 2 = 3 + 2 = 5 \)
\( a_3 = a_2 + 2 = 5 + 2 = 7 \)
\( a_4 = a_3 + 2 = 7 + 2 = 9 \)
\( a_5 = a_4 + 2 = 9 + 2 = 11 \)
Sequence: 3, 5, 7, 9, 11, ...
Fibonacci: \( a_1 = 1, a_2 = 1, \quad a_n = a_{n-1} + a_{n-2} \)
Find next 3 terms:
\( a_3 = a_2 + a_1 = 1 + 1 = 2 \)
\( a_4 = a_3 + a_2 = 2 + 1 = 3 \)
\( a_5 = a_4 + a_3 = 3 + 2 = 5 \)
Sequence: 1, 1, 2, 3, 5, ...
4. Explicit vs Recursive
| Aspect | Explicit Formula | Recursive Formula |
|---|---|---|
| Definition | Uses position \( n \) only | Uses previous term(s) |
| Example | \( a_n = 3n + 1 \) | \( a_1 = 4, a_n = a_{n-1} + 3 \) |
| Advantage | Find any term directly | Easy to describe pattern |
| Disadvantage | May be hard to find | Must calculate all previous terms |
5. Find a Recursive Formula
Steps:
1. Identify the first term \( a_1 \)
2. Find the relationship between consecutive terms
3. Write the recursive equation
4. State both the initial condition and recursive equation
Examples:
Find recursive formula: 5, 8, 11, 14, 17, ...
First term: \( a_1 = 5 \)
Pattern: Each term is 3 more than previous
Relationship: \( a_n = a_{n-1} + 3 \)
Recursive Formula: \( a_1 = 5, \quad a_n = a_{n-1} + 3 \)
Find recursive formula: 2, 6, 18, 54, ...
First term: \( a_1 = 2 \)
Pattern: Each term is 3 times the previous
Relationship: \( a_n = 3 \cdot a_{n-1} \)
Recursive Formula: \( a_1 = 2, \quad a_n = 3a_{n-1} \)
6. Convert Recursive to Explicit
For Arithmetic Sequences:
Recursive: \( a_1 = c, \quad a_n = a_{n-1} + d \)
Explicit: \( a_n = a_1 + (n-1)d \)
where \( a_1 \) = first term, \( d \) = common difference
For Geometric Sequences:
Recursive: \( a_1 = c, \quad a_n = r \cdot a_{n-1} \)
Explicit: \( a_n = a_1 \cdot r^{n-1} \)
where \( a_1 \) = first term, \( r \) = common ratio
Example:
Convert: \( a_1 = 5, \quad a_n = a_{n-1} + 3 \)
This is arithmetic with \( a_1 = 5 \) and \( d = 3 \)
Use formula: \( a_n = a_1 + (n-1)d \)
\( a_n = 5 + (n-1)(3) \)
\( a_n = 5 + 3n - 3 \)
Explicit: \( a_n = 3n + 2 \)
7. Convert Explicit to Recursive
Steps:
1. Find \( a_1 \) by substituting \( n = 1 \) into explicit formula
2. Find \( a_n \) and \( a_{n-1} \) using the explicit formula
3. Determine the relationship between them
4. Write recursive formula with initial condition
Examples:
Convert: \( a_n = 4n - 1 \)
Find first term: \( a_1 = 4(1) - 1 = 3 \)
Find pattern: \( a_n = 4n - 1 \) and \( a_{n-1} = 4(n-1) - 1 = 4n - 5 \)
Relationship: \( a_n - a_{n-1} = (4n-1) - (4n-5) = 4 \)
So: \( a_n = a_{n-1} + 4 \)
Recursive: \( a_1 = 3, \quad a_n = a_{n-1} + 4 \)
Convert: \( a_n = 3 \cdot 2^{n-1} \)
Find first term: \( a_1 = 3 \cdot 2^{1-1} = 3 \cdot 1 = 3 \)
This is geometric with ratio 2
\( \frac{a_n}{a_{n-1}} = \frac{3 \cdot 2^{n-1}}{3 \cdot 2^{n-2}} = 2 \)
So: \( a_n = 2 \cdot a_{n-1} \)
Recursive: \( a_1 = 3, \quad a_n = 2a_{n-1} \)
8. Common Sequence Types
Arithmetic Sequence:
Add the same number (common difference) each time
Recursive: \( a_n = a_{n-1} + d \)
Explicit: \( a_n = a_1 + (n-1)d \)
Example: 2, 5, 8, 11, 14, ... (d = 3)
Geometric Sequence:
Multiply by the same number (common ratio) each time
Recursive: \( a_n = r \cdot a_{n-1} \)
Explicit: \( a_n = a_1 \cdot r^{n-1} \)
Example: 3, 6, 12, 24, 48, ... (r = 2)
9. Quick Reference Summary
Key Formulas:
Explicit: \( a_n = f(n) \) - uses position only
Recursive: \( a_n = f(a_{n-1}) \) - uses previous term(s)
Arithmetic Explicit: \( a_n = a_1 + (n-1)d \)
Arithmetic Recursive: \( a_1 = c, \quad a_n = a_{n-1} + d \)
Geometric Explicit: \( a_n = a_1 \cdot r^{n-1} \)
Geometric Recursive: \( a_1 = c, \quad a_n = r \cdot a_{n-1} \)
📚 Study Tips
✓ Explicit formulas are faster for finding specific terms
✓ Recursive formulas always need initial condition(s)
✓ Arithmetic: add/subtract same value each time
✓ Geometric: multiply/divide by same value each time
✓ Check your work by calculating a few terms both ways