More on Sequences

On slide 6, we learned what sequences are; we promised to resume our discussion on them, which we do now.

Since a sequence is an ordered container of elements, it makes sense to refer to its terms by their position in the sequence. In fact, we can use variables to denote the terms: $a_1$ is the first (leftmost) term, $a_2$ is the next term, etc.

Some books begin counting at $0$, as in $a_0$. Also, many programming languages have arrays starting with the index of $0$. Also, it is common to use lowercase letters to name terms, e.g., $b_{12}$, $k_3$, or $x_{109}$.

Example: In the sequence $\langle 10, 11, 12, 13, \dots\rangle$, we denote the terms as $a_1 = 10$, $a_2 = 11$, $a_3 = 12$, $a_4 = 13$, etc. A shorthand for a sequence of $n$ terms is $(a_i)^n_{i=1} = $$\;(a_1, a_2, \dots, a_n) = $$\;\langle a_1, a_2, \dots, a_n\rangle$ (

$(a_i)^n_{i=1}$

). The variable $n$ usually stands for the position of the last term in the sequence but can also be replaced with $\infty$ for infinite sets.

As we've seen in our sequence examples so far, we can list the terms of a sequence (if it is finite and short) and use the ellipsis ($\dots$) to indicate continuation. Moreover, sequences with special term patterns have names, such as arithmetic sequences and geometric sequences; we'll define those in the next slides.