Sequences
Eleventh Grade Mathematics - Complete Notes & Formulas
📚 What is a Sequence?
A sequence is an ordered list of numbers following a specific pattern. Each number in the sequence is called a term.
General notation: \(a_1, a_2, a_3, ..., a_n\) where \(a_n\) represents the \(n\)th term.
📊 Arithmetic Sequences
Definition
A sequence where each term is obtained by adding a common difference (d) to the previous term.
Example: 2, 5, 8, 11, 14, ... (common difference \(d = 3\))
Key Formulas
Explicit Formula (nth term):
\[a_n = a_1 + (n-1)d\]
Where: \(a_1\) = first term, \(n\) = term position, \(d\) = common difference
Recursive Formula:
\[a_n = a_{n-1} + d, \quad n \geq 2\]
With initial condition: \(a_1\) = first term
Common Difference:
\[d = a_n - a_{n-1}\]
Sum of First n Terms:
\[S_n = \frac{n}{2}[2a_1 + (n-1)d]\]
\[S_n = \frac{n}{2}(a_1 + a_n)\]
Use first formula when last term unknown, second when last term \(a_n\) is known
📈 Geometric Sequences
Definition
A sequence where each term is obtained by multiplying the previous term by a common ratio (r).
Example: 3, 6, 12, 24, 48, ... (common ratio \(r = 2\))
Key Formulas
Explicit Formula (nth term):
\[a_n = a_1 \cdot r^{n-1}\]
Where: \(a_1\) = first term, \(n\) = term position, \(r\) = common ratio
Recursive Formula:
\[a_n = r \cdot a_{n-1}, \quad n \geq 2\]
With initial condition: \(a_1\) = first term
Common Ratio:
\[r = \frac{a_n}{a_{n-1}}\]
Sum of First n Terms (Finite):
When \(r \neq 1\):
\[S_n = \frac{a_1(1-r^n)}{1-r} \quad \text{or} \quad S_n = \frac{a_1(r^n-1)}{r-1}\]
When \(r = 1\):
\[S_n = n \cdot a_1\]
Sum of Infinite Terms:
Only converges when \(|r| < 1\):
\[S_\infty = \frac{a_1}{1-r}\]
If \(|r| \geq 1\), the series does not converge
🔢 Evaluating Formulas
Explicit Formula Evaluation
Substitute the term number \(n\) directly into the formula to find the value of that term.
Example: If \(a_n = 3n + 2\), then \(a_5 = 3(5) + 2 = 17\)
Recursive Formula Evaluation
Calculate terms sequentially, using the previous term to find the next term.
Example: If \(a_1 = 2\) and \(a_n = 2a_{n-1}\), then \(a_2 = 2(2) = 4\), \(a_3 = 2(4) = 8\), etc.
🏷️ Classifying Sequences and Formulas
How to Identify Sequence Type
✓ Arithmetic Sequence:
• Consecutive terms have a constant difference
• Check: \(a_2 - a_1 = a_3 - a_2 = d\)
✓ Geometric Sequence:
• Consecutive terms have a constant ratio
• Check: \(\frac{a_2}{a_1} = \frac{a_3}{a_2} = r\)
✓ Neither:
• No constant difference or ratio exists
Formula Classification
✓ Explicit Formula:
• Expresses \(a_n\) in terms of \(n\) only
• Can find any term directly without computing previous terms
✓ Recursive Formula:
• Expresses \(a_n\) in terms of previous term(s)
• Requires initial condition(s) and previous terms to find later terms
✍️ Writing Formulas
Steps for Arithmetic Sequence
For Explicit Formula:
1. Identify first term \(a_1\)
2. Find common difference \(d = a_2 - a_1\)
3. Substitute into \(a_n = a_1 + (n-1)d\)
For Recursive Formula:
1. State \(a_1 = \) first term
2. Find common difference \(d\)
3. Write \(a_n = a_{n-1} + d\) for \(n \geq 2\)
Steps for Geometric Sequence
For Explicit Formula:
1. Identify first term \(a_1\)
2. Find common ratio \(r = \frac{a_2}{a_1}\)
3. Substitute into \(a_n = a_1 \cdot r^{n-1}\)
For Recursive Formula:
1. State \(a_1 = \) first term
2. Find common ratio \(r\)
3. Write \(a_n = r \cdot a_{n-1}\) for \(n \geq 2\)
🔄 Converting Between Formulas
Recursive → Explicit (Arithmetic)
Given: \(a_1\) and \(a_n = a_{n-1} + d\)
Step 1: Identify \(a_1\) and \(d\) from recursive formula
Step 2: Substitute into explicit formula:
\[a_n = a_1 + (n-1)d\]
Explicit → Recursive (Arithmetic)
Given: \(a_n = a_1 + (n-1)d\)
Step 1: Identify \(a_1\) and \(d\) from explicit formula
Step 2: Write recursive formula:
\[a_1 = \text{(first term)}, \quad a_n = a_{n-1} + d\]
Recursive → Explicit (Geometric)
Given: \(a_1\) and \(a_n = r \cdot a_{n-1}\)
Step 1: Identify \(a_1\) and \(r\) from recursive formula
Step 2: Substitute into explicit formula:
\[a_n = a_1 \cdot r^{n-1}\]
Explicit → Recursive (Geometric)
Given: \(a_n = a_1 \cdot r^{n-1}\)
Step 1: Identify \(a_1\) and \(r\) from explicit formula
Step 2: Write recursive formula:
\[a_1 = \text{(first term)}, \quad a_n = r \cdot a_{n-1}\]
📋 Quick Reference Table
| Property | Arithmetic Sequence | Geometric Sequence |
|---|---|---|
| Pattern | Add constant difference | Multiply by constant ratio |
| Explicit Formula | \(a_n = a_1 + (n-1)d\) | \(a_n = a_1 \cdot r^{n-1}\) |
| Recursive Formula | \(a_n = a_{n-1} + d\) | \(a_n = r \cdot a_{n-1}\) |
| Key Parameter | Common difference \(d\) | Common ratio \(r\) |
| Sum Formula | \(S_n = \frac{n}{2}[2a_1 + (n-1)d]\) | \(S_n = \frac{a_1(1-r^n)}{1-r}\) |
💡 Key Points to Remember
✓ Arithmetic sequences have constant differences between terms
✓ Geometric sequences have constant ratios between terms
✓ Explicit formulas allow direct calculation of any term
✓ Recursive formulas require previous terms to find the next term
✓ Always identify the first term and common difference/ratio first
✓ Sum formulas help find the total of multiple terms efficiently
✓ Infinite geometric series only converge when |r| < 1
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