🔢 Sequences and Patterns Formulas
🎯 Arithmetic Sequences
General Term
$$a_n = a_1 + (n-1)d$$
Usage: Find the nth term of an arithmetic sequence Micro-example: Sequence 3, 7, 11, 15, … has $a_1 = 3$, $d = 4$, so $a_n = 3 + 4(n-1) = 4n - 1$
Sum of First n Terms
$$S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}[2a_1 + (n-1)d]$$
Usage: Find sum of first n terms of arithmetic sequence Micro-example: Sum of 2, 5, 8, 11, 14 is $S_5 = \frac{5}{2}(2 + 14) = 40$
🔢 Geometric Sequences
General Term
$$a_n = a_1 \cdot r^{n-1}$$
Usage: Find the nth term of a geometric sequence Micro-example: Sequence 2, 6, 18, 54, … has $a_1 = 2$, $r = 3$, so $a_n = 2 \cdot 3^{n-1}$
Sum of First n Terms
$$S_n = \frac{a_1(1-r^n)}{1-r} \text{ when } r \neq 1$$
Usage: Find sum of first n terms of geometric sequence Micro-example: Sum of 3, 6, 12, 24 is $S_4 = \frac{3(1-2^4)}{1-2} = \frac{3(-15)}{-1} = 45$
Infinite Geometric Series
$$S_\infty = \frac{a_1}{1-r} \text{ when } |r| < 1$$
Usage: Find sum of infinite geometric series Micro-example: Sum of $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + …$ is $S_\infty = \frac{1}{1-\frac{1}{2}} = 2$
🎨 Special Sequences
Fibonacci Sequence
$$F_n = F_{n-1} + F_{n-2}$$
Where: $F_1 = 1$, $F_2 = 1$ Usage: Generate Fibonacci numbers Micro-example: $F_3 = F_2 + F_1 = 1 + 1 = 2$, $F_4 = F_3 + F_2 = 2 + 1 = 3$
Triangular Numbers
$$T_n = \frac{n(n+1)}{2}$$
Usage: Find nth triangular number Micro-example: $T_4 = \frac{4 \cdot 5}{2} = 10$
Square Numbers
$$S_n = n^2$$
Usage: Find nth square number Micro-example: $S_5 = 5^2 = 25$
Cubic Numbers
$$C_n = n^3$$
Usage: Find nth cubic number Micro-example: $C_3 = 3^3 = 27$
🔍 Pattern Recognition
Linear Pattern
$$a_n = an + b$$
Usage: When first differences are constant Micro-example: Sequence 2, 5, 8, 11, … has $a_n = 3n - 1$
Quadratic Pattern
$$a_n = an^2 + bn + c$$
Usage: When second differences are constant Micro-example: Sequence 1, 4, 9, 16, 25, … has $a_n = n^2$
Exponential Pattern
$$a_n = a \cdot r^n$$
Usage: When ratios are constant Micro-example: Sequence 2, 4, 8, 16, 32, … has $a_n = 2^n$
💡 Quick Reference
| Sequence Type | General Term | Sum Formula |
|---|---|---|
| Arithmetic | $a_n = a_1 + (n-1)d$ | $S_n = \frac{n}{2}(a_1 + a_n)$ |
| Geometric | $a_n = a_1 \cdot r^{n-1}$ | $S_n = \frac{a_1(1-r^n)}{1-r}$ |
| Fibonacci | $F_n = F_{n-1} + F_{n-2}$ | No simple formula |
| Triangular | $T_n = \frac{n(n+1)}{2}$ | $S_n = \frac{n(n+1)(n+2)}{6}$ |
| Square | $S_n = n^2$ | $S_n = \frac{n(n+1)(2n+1)}{6}$ |
| Cubic | $C_n = n^3$ | $S_n = \left[\frac{n(n+1)}{2}\right]^2$ |
⚠️ Common Pitfalls
- Off-by-one errors: Remember $n-1$ in formulas
- Sign errors: Check signs in difference calculations
- Wrong sequence type: Identify arithmetic vs geometric
- Formula confusion: Use correct sum formula
- Infinite series: Only works when $|r| < 1$