Sequences

A sequence is a list of numbers: like 4,9,3,2,17.

An arithmetic sequence is a list where each number is generated by adding a constant to the previous number. An example is 10,13,16,19,22,25. In this example, the first term ( $t_1$ ) is 10, and the “common difference” ( $d$ )—that is, the difference between any two adjacent numbers—is 3. Another example is 25,22,19,16,13,10. In this example $t_1=25$ , and $d=\left(\mathrm{–3}\right)$ . In both of these examples, $n$ (the number of terms) is 6.

A geometric sequence is a list where each number is generated by multiplying a constant by the previous number. An example is 2,6,18,54,162. In this example, $t_1=2$ , and the “common ratio” ( $r$ )—that is, the ratio between any two adjacent numbers—is 3. Another example is 162,54,18,6,2. In this example $t_1=162$ , and $r=\frac{1}{3}$ . In both examples $n=5$ .

A recursive definition of a sequence means that you define each term based on the previous. So the recursive definition of an arithmetic sequence is $t_n=t_{n-1}+d$ , and the recursive definition of a geometric sequence is $t_n=r{t}_{n-1}$ .

An explicit definition of an arithmetic sequence means you define the $n^{th}$ term without making reference to the previous term. This is more useful, because it means you can find (for instance) the 20th term without finding all the other terms in between.

To find the explicit definition of an arithmetic sequence, you just start writing out the terms. The first term is always $t_1$ . The second term goes up by $d$ so it is $t_1+d$ . The third term goes up by $d$ again, so it is $(t_1+d)+d$ , or in other words, $t_1+2d$ . So we get a chart like this.

…and so on. From this you can see the generalization that $t_n=t_1+(n-1)d$ , which is the explicit definition we were looking for.

The explicit definition of a geometric sequence is arrived at the same way. The first term is $t_1$ ; the second term is $r$ times that, or $t_{1}r$ ; the third term is $r$ times that , or $t_1{r}^2$ ; and so on. So the general rule is $t_n=t_1{r}^{n-1}$ . Read this as: “ $t_1$ multiplied by $r$ , $(n–1)$ times.”