🔢 Sequences and Patterns Topics#
🎯 Core Subtopics#
Arithmetic Sequences#
- Definition: Each term differs by a constant amount
- General Form: $a_n = a_1 + (n-1)d$
- Sum Formula: $S_n = \frac{n}{2}(a_1 + a_n)$
- Micro-example: Sequence 3, 7, 11, 15, … has $d = 4$ and $a_n = 3 + 4(n-1)$
- Trap: Forgetting to subtract 1 in the formula
Geometric Sequences#
- Definition: Each term is multiplied by a constant ratio
- General Form: $a_n = a_1 \cdot r^{n-1}$
- Sum Formula: $S_n = \frac{a_1(1-r^n)}{1-r}$ when $r \neq 1$
- Micro-example: Sequence 2, 6, 18, 54, … has $r = 3$ and $a_n = 2 \cdot 3^{n-1}$
- Trap: Using wrong formula when $r = 1$
Pattern Recognition#
- Linear Patterns: $a_n = an + b$
- Quadratic Patterns: $a_n = an^2 + bn + c$
- Exponential Patterns: $a_n = a \cdot r^n$
- Micro-example: Sequence 1, 4, 9, 16, … follows $a_n = n^2$
- Trap: Assuming linear when it’s quadratic
Series Sums#
- Arithmetic Series: $S_n = \frac{n}{2}(a_1 + a_n)$
- Geometric Series: $S_n = \frac{a_1(1-r^n)}{1-r}$
- Infinite Geometric: $S_\infty = \frac{a_1}{1-r}$ when $|r| < 1$
- Micro-example: Sum of 1, 3, 5, 7, 9 is $S_5 = \frac{5}{2}(1 + 9) = 25$
- Trap: Using infinite series formula when series is finite
Special Sequences#
- Fibonacci: $F_n = F_{n-1} + F_{n-2}$ with $F_1 = F_2 = 1$
- Triangular Numbers: $T_n = \frac{n(n+1)}{2}$
- Square Numbers: $S_n = n^2$
- Cubic Numbers: $C_n = n^3$
- Micro-example: First 5 triangular numbers: 1, 3, 6, 10, 15
- Trap: Confusing triangular and square numbers
Recursive Sequences#
- Definition: Each term depends on previous terms
- Common Types: Fibonacci, arithmetic, geometric
- Micro-example: $a_n = 2a_{n-1} + 1$ with $a_1 = 1$ gives 1, 3, 7, 15, …
- Trap: Not finding the closed form when possible
🚨 Common Traps#
- Formula Confusion: Mixing up arithmetic and geometric formulas
- Off-by-One Errors: Forgetting to subtract 1 in $n-1$
- Sign Errors: Wrong signs in difference calculations
- Pattern Assumptions: Assuming linear when it’s quadratic
- Series vs Sequence: Confusing sum formulas
💡 Quick Tips#
- Arithmetic: Look for constant differences
- Geometric: Look for constant ratios
- Patterns: Check differences, then ratios
- Sums: Use appropriate formula for sequence type
- Special: Memorize common special sequences