Sequences

Sequences are ordered lists of numbers arranged according to a specific rule or pattern. The nth term is found using a formula based on its position.

For example, the height of a bouncing ball decreases with each bounce, forming a decreasing sequence in which each height is a fixed fraction of the previous one.

Each number in a sequence is called a term, and the position of the ordered terms determines its value.

When the pattern is clear, dots indicate the sequence continues.

Some sequences define terms using previous ones, called recursive sequences. For example, the nth term is defined using the (n-1)th term.

For instance, in the Fibonacci sequence, each term equals the sum of the two preceding terms.

Partial sums are the sums of the first few terms of a sequence. Often shown using sigma notation, partial sums help analyze how the sum grows as more terms are added.

Each of these is called a partial sum: S₁ is the first, S₂ is the second, and S_n is the nth’s term summation.

The sequence formed by these is called the sequence of partial sums.

For example, each week's deposit is a term when tracking weekly savings. Partial sums show how total savings grow over time.