A sequence is an ordered list of numbers that follows a specific pattern or rule. Each number in the sequence is called a term. For example, the sequence 2, 4, 6, 8, 10 follows the pattern where each term equals 2 times its position number.
There are several types of sequences. Arithmetic sequences have a constant difference between consecutive terms, like 3, 7, 11, 15 where we add 4 each time. Geometric sequences have a constant ratio, like 2, 6, 18, 54 where we multiply by 3. Fibonacci sequences add the two previous terms, like 1, 1, 2, 3.
An arithmetic sequence has a constant difference between consecutive terms. For example, in the sequence 5, 8, 11, 14, 17, 20, the common difference is 3. The general formula is a sub n equals a sub 1 plus n minus 1 times d, where d is the common difference.
A geometric sequence has a constant ratio between consecutive terms. For example, in the sequence 2, 6, 18, 54, 162, the common ratio is 3. The general formula is a sub n equals a sub 1 times r to the power of n minus 1, where r is the common ratio. Notice how geometric sequences grow much faster than arithmetic sequences.
Sequences have many real-world applications. In finance, compound interest follows geometric sequences. Population growth can be modeled using sequences. Computer algorithms often use sequences for sorting and searching. Natural patterns like flower petals and shell spirals follow the Fibonacci sequence. Even musical scales are based on geometric sequences of frequencies.