Arithmetic and Geometric Sequences

Arithmetic Sequences

Each term is found by adding a constant called the common difference $d$.

$a_n = a_1 + (n-1)d$

Example: $3, 7, 11, 15, 19, ...$

• $a_1 = 3$, $d = 4$
• $a_n = 3 + (n-1)(4) = 4n - 1$
• $a_{10} = 4(10) - 1 = 39$

Geometric Sequences

Each term is found by multiplying by a constant called the common ratio $r$.

$a_n = a_1 \cdot r^{n-1}$

Example: $2, 6, 18, 54, ...$

• $a_1 = 2$, $r = 3$
• $a_n = 2 \cdot 3^{n-1}$
• $a_5 = 2 \cdot 3^4 = 2 \cdot 81 = 162$

Identifying the Type

| Sequence | Differences | Ratios | Type | |----------|------------|--------|------| | 5, 8, 11, 14 | 3, 3, 3 | — | Arithmetic ($d = 3$) | | 3, 6, 12, 24 | 3, 6, 12 | 2, 2, 2 | Geometric ($r = 2$) | | 1, 4, 9, 16 | 3, 5, 7 | — | Neither (perfect squares) |

Arithmetic Series (Sum)

$S_n = \frac{n}{2}(a_1 + a_n)$

Sum of first 20 terms of $3, 7, 11, ...$: $a_{20} = 3 + 19(4) = 79$ $S_{20} = \frac{20}{2}(3 + 79) = 10(82) = 820$

Connection to Functions

• Arithmetic sequences → Linear functions ($f(x) = mx + b$)
• Geometric sequences → Exponential functions ($f(x) = ab^x$)

Quick check: Subtract consecutive terms. If the differences are constant → arithmetic. Divide consecutive terms. If the ratios are constant → geometric.