A sequence is an ordered list of numbers that follows a specific pattern or rule. For example, the sequence two, four, six, eight, ten represents the even numbers. Each number in the sequence is called a term, and we can see the pattern is adding two to get the next term.
A series is the sum of the terms of a sequence. When we take our sequence of even numbers two, four, six, eight, ten and add them together, we get a series. The sum two plus four plus six plus eight plus ten equals thirty. This transformation from sequence to series is fundamental in mathematics.
Arithmetic sequences are sequences where there is a constant difference between consecutive terms. For example, in the sequence three, seven, eleven, fifteen, nineteen, the common difference is four. We can find any term using the formula a sub n equals a sub one plus n minus one times d, where d is the common difference.
Geometric sequences are sequences where there is a constant ratio between consecutive terms. For example, in the sequence two, six, eighteen, fifty-four, one hundred sixty-two, the common ratio is three. Each term is found by multiplying the previous term by three. The formula for the nth term is a sub n equals a sub one times r to the power of n minus one.
To summarize what we have learned: sequences are ordered lists of numbers that follow specific patterns, while series are the sums of sequence terms. Arithmetic sequences have constant differences, geometric sequences have constant ratios, and both types have useful formulas for finding terms and calculating sums.