Understanding Sequences

Sequences are ordered lists of numbers that follow specific patterns. The two most common types are arithmetic sequences and geometric sequences.

In an arithmetic sequence, each term differs from the previous by a constant value called the common difference $d$. For example, 35, 32, 29, 26,... is arithmetic with d = -3. To find terms, use the explicit formula: $a_n = a_1 + (n-1)d$ or the recursive formula: $a_n = a_{n-1} + d$.

In a geometric sequence, each term is multiplied by a constant value called the common ratio $r$. For instance, 1, -6, 36, -216,... is geometric with r = -6. The explicit formula is $a_n = a_1 \cdot r^{n-1}$ and the recursive formula is $a_n = a_{n-1} \cdot r$.

💡 When identifying sequences, check if subtracting consecutive terms gives you the same number (arithmetic) or if dividing consecutive terms gives you the same number (geometric). If neither works, it's probably neither type!

Some sequences don't fit either pattern. For example, 1, 8, 27, 64, 125,... is neither arithmetic nor geometric—it's actually $a_n = n^3$, the sequence of perfect cubes.