Sequences as Discrete Functions

We write the terms of a sequence as $t_1,t_2,t_3,\dots ,t_n,\dots$. The subscript is the term NUMBER (which position in the list), and $t_n$ is the term VALUE (what's actually there). For the sequence $5,8,11,14,\dots$ we'd say $t_1=5$, $t_2=8$, $t_3=11$, and so on. The first term is often also called $a$. Because each position gives exactly one value, a sequence really is a function: you could write $f(n)$ instead of $t_n$, and its graph is the set of ordered pairs $(n,t_n)$, such as $(1,5),(2,8),(3,11)$. A sequence that stops, like the 12 monthly balances in a year, is finite; one that goes on forever, shown by the trailing "$\dots$", is infinite.

There are two completely different ways to describe a sequence with a formula. An EXPLICIT formula gives you $t_n$ directly from $n$ — plug in $n=50$ and you instantly get $t_{50}$ with no other work. A RECURSIVE formula instead tells you how to get the next term from the previous one(s), so you must know a starting value and climb your way up term by term.

Why bother with recursion if explicit formulas are faster? Because recursion describes the process, and many real situations are processes: "each year the balance is 1.05 times last year's balance plus a 100 dollar deposit" is naturally $t_n=1.05t_{n-1}+100$. Recursion is also how spreadsheets and computer programs build sequences, one cell from the one above. The two sequence families in this unit have simple recursive forms that convert straight to explicit ones: an arithmetic rule $t_n=t_{n-1}+d$ gives $t_n=a+(n-1)d$, and a geometric rule $t_n=r{t}_{n-1}$ gives $t_n=a{r}^{n-1}$. Some recursions use two earlier terms, like the Fibonacci sequence $t_1=t_2=1$, $t_n=t_{n-1}+t_{n-2}$, which gives $1,1,2,3,5,8,13,\dots$.

Explicit and recursive definitions of sequences | Precalculus | Khan Academy

1. Write out at least 4–5 consecutive terms.
2. Compute first differences: $t_2-t_1,t_3-t_2,t_4-t_3,\dots$
3. If the first differences are constant, you likely have an arithmetic sequence (next topic).
4. If instead consecutive terms have a constant RATIO, you likely have a geometric sequence.
5. If neither is constant, check second differences (differences of the differences) — a constant second difference signals a quadratic-type pattern, which is common but not arithmetic or geometric.

When the terms are not arithmetic or geometric, compare each term to its position number $n$ and look for a familiar pattern. Squares give $1,4,9,16=n^2$; one less than the squares gives $0,3,8,15=n^2-1$. Fractions often split into a numerator pattern and a denominator pattern: $\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5}$ is $t_n=\frac{n}{n+1}$. Alternating signs come from powers of $-1$: $1,-1,1,-1$ is ${(-1)}^{n+1}$, so $3,-6,9,-12$ is $t_n={(-1)}^{n+1}(3n)$. Always test your formula on at least two terms that you did not use to build it.