Patterns and Sequences

A sequence is a set of numbers arranged in a definite order.

• Each number in a sequence is called a term.

• A sequence may be written using symbols

  t₁,t₂,t₃,…,t_n
• The general term of a sequence is denoted by t_n.​

Given Sequence: \[ 1, 4, 7, 10, \ldots \]

Step 1: Find the difference between consecutive terms

\[ 4 - 1 = 3 \]

\[ 7 - 4 = 3 \]

\[ 10 - 7 = 3 \]

Step 2: Observe the pattern

The difference between any term and its preceding term is constant \[ (+3). \]

Given Sequence: \[ 4, 16, 64, \ldots \]

Step 1: Find the ratio between consecutive terms

\[ \frac{16}{4} = 4 \]

\[ \frac{64}{16} = 4 \]

Step 2: Observe the pattern

Each term is obtained by multiplying the preceding term by a constant factor \[ (\times 4). \]

• A sequence is an ordered list of numbers following a rule.
• Each number in a sequence is called a term.
• \[ a_n \] or \[ t_n \] represents the nth term.
• A finite sequence has a last term; an infinite sequence does not.