🔢 Sequences and Patterns Reference
🎯 Key Concepts
Sequence
An ordered list of numbers, often following a pattern.
Term
An individual number in a sequence.
Arithmetic Sequence
A sequence where each term is obtained by adding a constant difference.
Geometric Sequence
A sequence where each term is obtained by multiplying by a constant ratio.
Series
The sum of terms in a sequence.
📐 Arithmetic Sequences
General Form
$$a_n = a_1 + (n-1)d$$
Where:
- $a_n$ = nth term
- $a_1$ = first term
- $d$ = common difference
- $n$ = term number
Example: Sequence 2, 5, 8, 11, … has $a_1 = 2$ and $d = 3$
Sum of Arithmetic Series
$$S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}[2a_1 + (n-1)d]$$
Usage: Find sum of first n terms Example: Sum of 2, 5, 8, 11, 14 is $S_5 = \frac{5}{2}(2 + 14) = 40$
🔢 Geometric Sequences
General Form
$$a_n = a_1 \cdot r^{n-1}$$
Where:
- $a_n$ = nth term
- $a_1$ = first term
- $r$ = common ratio
- $n$ = term number
Example: Sequence 3, 6, 12, 24, … has $a_1 = 3$ and $r = 2$
Sum of Geometric Series
$$S_n = \frac{a_1(1-r^n)}{1-r} \text{ when } r \neq 1$$
Usage: Find sum of first n terms Example: Sum of 3, 6, 12, 24 is $S_4 = \frac{3(1-2^4)}{1-2} = \frac{3(-15)}{-1} = 45$
🎨 Special Sequences
Fibonacci Sequence
$$F_n = F_{n-1} + F_{n-2}$$
Where: $F_1 = 1$, $F_2 = 1$ Example: 1, 1, 2, 3, 5, 8, 13, 21, …
Triangular Numbers
$$T_n = \frac{n(n+1)}{2}$$
Example: $T_4 = \frac{4 \cdot 5}{2} = 10$ (1, 3, 6, 10, …)
Square Numbers
$$S_n = n^2$$
Example: 1, 4, 9, 16, 25, …
Cubic Numbers
$$C_n = n^3$$
Example: 1, 8, 27, 64, 125, …
🔍 Pattern Recognition
Common Patterns
- Linear: $a_n = an + b$
- Quadratic: $a_n = an^2 + bn + c$
- Exponential: $a_n = a \cdot r^n$
- Recursive: $a_n = f(a_{n-1}, a_{n-2}, …)$
Finding the Pattern
- Look for differences between consecutive terms
- Check for ratios between consecutive terms
- Try polynomial formulas if differences are constant
- Look for recursive relationships
💡 Problem-Solving Strategies
- Identify the type of sequence (arithmetic, geometric, other)
- Find the pattern by examining differences or ratios
- Write the general formula for the nth term
- Use appropriate formulas for sums
- Check your work by verifying a few terms
⚠️ Common Mistakes
- Confusing arithmetic and geometric sequences
- Wrong formula for series sums
- Off-by-one errors in term numbers
- Sign errors in difference calculations
- Not recognizing special sequence types