A sequence is one of the simplest ways to describe a pattern in mathematics: you list numbers in an order, and each number has a position.
- Because a sequence is ordered, the position matters.
- We label terms using subscripts:
- $u_1$ is the first term
- $u_2$ is the second term
- $u_3$ is the third term
- in general, $u_n$ is the $n$th term
- Here, the letter $u$ names the sequence, and the subscript tells you the term number.
The sequence $$1, 3, 5, 7, 9, \ldots$$ has a clear first term, second term, third term, and so on.
- Using $u_1, u_2, u_3, \ldots$ is more useful than labels like $a,b,c,\ldots$ because the subscript directly shows the position.
- This matters when you want, for example, "the 20th term".
- Different subjects and contexts sometimes start sequences at $u_0$ instead of $u_1$ (especially in computer science).
- Always check what "first term" means in the question.
- An explicit (position-to-term) formula lets you calculate any term without calculating all the ones before it.
- For the odd numbers, $$u_n = 2n - 1 \quad \text{for } n \ge 1$$
- Check it quickly:
- $u_1 = 2(1)-1 = 1$
- $u_2 = 2(2)-1 = 3$
- $u_5 = 2(5)-1 = 9$
- A key idea is that in sequences, $n$ is a term number, so it must be a whole number.
In a sequence, $n$ represents a position (1st, 2nd, 3rd, …), so you do not substitute values like $n=2.5$.
- Many sequences in this course are linear sequences, meaning their explicit formula has the form: $$u_n = an + b$$
- This looks like a linear function $y = ax + b$, but there is a difference:
- A linear function is defined for all real $x$ (often continuous)
- A linear sequence is defined only at integer inputs $n = 1,2,3,\ldots$ (discrete points)
- A linear function is like a continuous line you can draw without lifting your pen.
- A linear sequence is like the same line, but only the dots at $x=1,2,3,\ldots$ are "allowed".
A recursive rule tells you how to go from one term to the next. It must include a starting value.
$$u_{n+1} = u_n + 7, \quad u_1 = -4$$
To generate terms, you repeatedly apply the rule:
- $u_1=-4$
- $u_2=u_1+7=-4+7=3$
- $u_3=u_2+7=3+7=10$
- $u_4=17$
- $u_5=24$
This process is sometimes called iteration.
- When you see $u_{n+1}$, read it as "the next term".
- When you see $u_n$, read it as "the current term".
1. Add/subtract a constant (often produces a linear sequence)
2. Multiply then add/subtract (often produces geometric-like growth)
$$u_{n+1}=4u_n-1$$
$$u_{n+1}=2u_n+1$$
These tend to grow quickly because each step depends on multiplication.
If the difference is constant, the sequence is linear.
$$1,3,5,7,9,\ldots$$
Compute differences:
Constant difference of 2 means the sequence is linear.
- Suppose you are given a linear sequence and want a direct rule.
- If the first term is $u_1$ and the common difference is $d$, then $$u_n = u_1 + (n-1)d$$
- For $1,3,5,7,\ldots$ we have $u_1=1$ and $d=2$: $$u_n = 1 + (n-1)\cdot 2 = 2n-1$$
- In exams, first compute a few first differences.
- If they are constant, you can usually write $u_n=u_1+(n-1)d$ immediately.
- A lot of sequence questions begin with: "Find the first five terms…".
- The method is always the same.
- Write down the starting term (usually $u_1$).
- Substitute into the recursive rule to get $u_2$.
- Repeat to get $u_3, u_4, \ldots$.
Find the first five terms of the sequence $u_{n+1}=\frac{1}{2}u_n+1$ with $u_1=3$.
Solution
$$u_1=3$$
$$u_2=\frac{1}{2}(3)+1=\frac{3}{2}+1=\frac{5}{2}$$
$$u_3=\frac{1}{2}\left(\frac{5}{2}\right)+1=\frac{5}{4}+1=\frac{9}{4}$$
$$u_4=\frac{1}{2}\left(\frac{9}{4}\right)+1=\frac{9}{8}+1=\frac{17}{8}$$
$$u_5=\frac{1}{2}\left(\frac{17}{8}\right)+1=\frac{17}{16}+1=\frac{33}{16}$$
So the first five terms are: $$3,\ \frac{5}{2},\ \frac{9}{4},\ \frac{17}{8},\ \frac{33}{16},\ldots$$
- A common mistake is to plug $n=1,2,3$ into a recursive formula as if it were explicit.
- In a recursive rule, you use the previous term value, not the index.
- Some sequences are easy to identify (like counting numbers or odd numbers).
- Others may look patternless at first.
- It is often useful to list several terms before deciding what is going on.
- In fact, even a list that looks random can still be generated by a rule, but the rule might not be easy to guess from only a few terms.
- Sometimes you are asked to find a term with a particular value, or the largest term below a limit.
- With recursive sequences, a practical strategy is to generate terms until you reach (or pass) the target.
A sequence is given by $u_{n+1}=2u_n-1$ and $u_1=2$. Find the term that has value 257.
Solution
Generate terms:
$$u_1=2$$
$$u_2=2(2)-1=3$$
$$u_3=2(3)-1=5$$
$$u_4=2(5)-1=9$$
You can spot a pattern: each time you double and subtract 1.
Continue until you reach 257: $$2,3,5,9,17,33,65,129,257$$
Count terms: this is the 9th term, so $u_9=257$.
For "largest term less than …", keep generating terms until you pass the limit, then take the term just before it.
- In the sequence $1,3,5,7,9,\ldots$, what is $u_4$?
- If $u_n=2n-1$, what is $u_{10}$?
- If $u_{n+1}=un+7$ and $u_1=-4$, what is $u_3$?
- Explain (in one sentence) the difference between an explicit formula and a recursive formula.