Number sequences

2, 4, 6, 8, ... everyone sees at once how it continues. Ordered rows of numbers like this are called sequences, and the individual numbers in them are their terms. It gets interesting when you need not the next term but the hundredth. Counting on will not get you there, but a formula will.

Always adding the same: arithmetic

In an arithmetic sequence the same number is added from term to term. In 3, 7, 11, 15, ... it is $+4$ each time. This fixed step is called the difference $d$. It can also be negative: in 20, 17, 14, ... we have $d = -3$, the sequence falls.

The formula for the terms

To jump straight to the n-th term, you start at the first term $a_1$ and add the difference once for every step you need. From the first to the n-th that is exactly $n - 1$ steps. This gives the formula for the terms.

Always multiplying the same: geometric

In a geometric sequence you don't add, you multiply. In 2, 6, 18, 54, ... each term is taken times 3. This fixed factor is called $q$. You find it by dividing a term by the one before it, here $6 \div 2 = 3$. Such sequences grow far faster than arithmetic ones, they practically explode as soon as $q$ is greater than 1.