Recognize Arithmetic Sequences by a Constant Difference
- An arithmetic sequence is one of the simplest ways to model "what comes next" in a list of numbers.
- The defining feature is that each term changes by the same amount as you move forward through the sequence.
Sequence
An ordered list of numbers where each number in the list is called a term.
Arithmetic sequence
A sequence in which the difference between consecutive terms is constant.
That constant difference is called the common difference, usually written as $d$.
Common difference
The constant value $d$ such that $u_{n+1}-u_n=d$ for every term in an arithmetic sequence.
- If $d>0$, the sequence is increasing.
- If $d<0$, it is decreasing.
- If $d=0$, the sequence is constant (all terms are the same).
Decide whether each sequence is arithmetic:
- $17, 19, 21, 23, 25, \ldots$ has differences $+2$ each time, so it is arithmetic with $d=2$.
- $64, 56, 48, 40, 32, \ldots$ has differences $-8$ each time, so it is arithmetic with $d=-8$.
- $1, 4, 9, 16, 25, \ldots$ has differences $3, 5, 7, 9, \ldots$ (not constant), so it is not arithmetic.
- Do not confuse a constant difference (arithmetic) with a constant ratio (geometric).
- For example, $2, 4, 8, 16, \ldots$ has a constant ratio of 2, but the differences are not constant.
Describe Arithmetic Sequences Recursively (Term-to-Term)
- A recursive formula (also called a term-to-term rule) tells you how to get the next term from the previous one.
- For an arithmetic sequence, the recursive form is: $$u_{n+1}=u_n+d$$
- To use this rule, you must also know a starting value, typically the first term $u_1$.
Given $a_{n+1}=a_n+3$ and $a_1=4$:
$$a_1=4$$
$$a_2=4+3=7$$
$$a_3=7+3=10$$
$$a_4=10+3=13$$
So the sequence begins $4, 7, 10, 13, \ldots$ and is arithmetic with $d=3$.
Given $b_{n+1}=b_n-2$ and $b_1=5$:
The sequence begins $5, 3, 1, -1, -3, \ldots$ and is arithmetic with $d=-2$.
When you see "$+\text{(constant)}$" or "$-\text{(constant)}$" in a recursion, it is a strong sign the sequence is arithmetic.
Write the Explicit Formula (Position-to-Term)
- An explicit formula (also called a position-to-term rule) allows you to calculate $u_n$ directly from $n$, without listing all earlier terms.
- For an arithmetic sequence with first term $u_1$ and common difference $d$, the explicit formula is: $$u_n=u_1+(n-1)d$$
- The factor $(n-1)$ appears because to get from the first term to the $n$th term, you take $(n-1)$ equal "steps", each of size $d$.
- Think of a staircase: $u_1$ is the first step.
- To reach step $n$, you move $(n-1)$ steps.
- Each step has the same height $d$ (up if $d>0$, down if $d<0$).
An arithmetic sequence has $u_1=7$ and $d=-3$. $$u_n=7+(n-1)(-3)=7-3n+3=10-3n$$
So $u_5=10-3(5)=-5$.
A common mistake is writing $u_n=u_1+nd$. Always check with $n=1$:
- Correct: $u_1+(1-1)d=u_1$.
- Incorrect: $u_1+1\cdot d=u_1+d$ (this gives the second term when $n=1$).
Find Missing Information from Given Terms
In many problems you are given some terms and must find the common difference $d$, the first term $u_1$, or another term.
Use Differences to Find $d$
- If you know two consecutive terms $u_k$ and $u_{k+1}$, then $$d=u_{k+1}-u_k$$
- If the terms are not consecutive, use the explicit formula with the term numbers.
Use Simultaneous Equations When Needed
If you are given two terms with different indices, substitute into $un=u1+(n-1)d$ to form two equations.
- An arithmetic sequence has $u_4=253$ and $u_5=291$.
- These are consecutive, so $$d=u_5-u_4=291-253=38$$
- Now use $u_4=u_1+3d$: $$253=u_1+3(38)=u_1+114$$
- So $u_1=139$.
- An arithmetic sequence has $u_3=31$ and $u_6=52$.
- Write equations: $$u_3 = u+1+2d=31 $$ $$u_6 = u_1+5d=52$$
- Subtract the first from the second: $$3d=21 \Rightarrow d=7$$
- Then $u_1+2(7)=31 \Rightarrow u_1=17$.
- Now find $u_{10}$: $$u_{10}=u_1+9d=17+63=80$$
- If $u_1=-2$ and $d=5$, what is $u_6$?
- If $u_2=11$ and $u_7=36$, what is $d$?
- The sequence is $3, 3, 3, 3, \ldots$. Is it arithmetic? What is $d$?
Solve Problems about Term Number (Membership)
- A frequent task is to decide whether a given value occurs in the sequence, and if it does, to find the term number $n$.
- Start with the explicit formula: $$u_n=u_1+(n-1)d$$
- Set $u_n$ equal to the given value and solve for $n$.
- The arithmetic sequence begins $7, 25, 43, \ldots$.
- First find the common difference: $$d=25-7=18$$
- So $$u_n=7+(n-1)18=7+18n-18=18n-11$$
- Find the term number of 367: $$18n-11=367 \Rightarrow 18n=378 \Rightarrow n=21$$
- So 367 is the 21st term.
- After solving for $n$, check that $n$ is a positive integer.
- If $n$ is not a whole number, the value is not a term of the arithmetic sequence.
Connect Arithmetic Sequences to Linear Functions
- The explicit formula $u_n=u_1+(n-1)d$ is linear in $n$.
- That means if you plot the points $(n,u_n)$ on a coordinate grid, they lie on a straight line.
- The slope of the line is the common difference $d$.
- Each time $n$ increases by 1, $u_n$ increases (or decreases) by $d$.
- This is why arithmetic sequences often appear in real contexts involving constant change, such as saving the same amount each week, or distance increasing by the same amount each second at constant speed.
- Arithmetic sequences are also called arithmetic progressions.
- The two names mean the same thing.
