Summary of Introduction to Numerical Sequences
Introduction to Numerical Sequences: Rules & Examples
Introduction
Mathematical sequences are ordered lists of numbers that follow a rule. Sequences help us model repeating patterns, predict values, and describe situations in finance, science, and computing.
A sequence is a list of numbers arranged in a specific order where each number is called a term.
## Term-to-term Rule (Difference between consecutive terms)
Term-to-term rules tell you how to get the next term from the previous term.
Examples
- Start with $7$ and add $3\tfrac{1}{2}$ each time:
$$7 + 3\tfrac{1}{2} = 10\tfrac{1}{2}$$ $$10\tfrac{1}{2} + 3\tfrac{1}{2} = 14$$ $$14 + 3\tfrac{1}{2} = 17\tfrac{1}{2}$$
The term-to-term rule is: add $3\tfrac{1}{2}$. The next two terms are:
$$17\tfrac{1}{2} + 3\tfrac{1}{2} = 21$$ $$21 + 3\tfrac{1}{2} = 24\tfrac{1}{2}$$
- Start with $10$ and subtract $0.2$ each time:
$$10 - 0.2 = 9.8$$ $$9.8 - 0.2 = 9.6$$ $$9.6 - 0.2 = 9.4$$
The term-to-term rule is: subtract $0.2$. The next two terms are:
$$9.4 - 0.2 = 9.2$$ $$9.2 - 0.2 = 9.0$$
When to use term-to-term
- When you know one term and want the next
- When the change between terms is constant (arithmetic sequences)
An arithmetic sequence is a sequence where the difference between consecutive terms is constant.
## Position-to-term Rule (n-th term formula)
A position-to-term rule gives the formula for the $n^{\text{th}}$ term directly from the position number $n$.
Example: linear decreasing sequence
Sequence: $18,\ 16,\ 14,\ 12,\ 10,\ \dots$
Observe the change: each term decreases by $2$. The position-to-term rule is:
$$\text{Term}_n = -2n + 20$$
Check: for $n = 1$ we get $-2(1) + 20 = 18$, for $n = 5$ we get $-2(5) + 20 = 10$, for $n = 21$ we get $-2(21) + 20 = -22$.
The position-to-term rule (or explicit formula) gives the $n^{\text{th}}$ term as a function of $n$.
How to find a position-to-term rule for arithmetic sequences
- Find the common difference $d$ between terms.
- Use the form $$\text{Term}_n = dn + c$$ or $$\text{Term}_n = a_1 + (n-1)d$$ and solve for $c$ or $a_1$ using a known term.
## Using Algebraic Rules (Function Machines)
Think of a function machine that takes an input $n$ (position) and gives an output (term). Use algebra to express that rule.
Example: If a machine does "multiply by 3 and add 1" then the position-to-term rule is:
$$\text{Term}_n = 3n + 1$$
This means when $n = 1$ you get $4$, when $n = 2$ you get $7$, and so on.
A function machine is a way to represent an algebraic rule that maps inputs to outputs.
Comparing Term-to-term and Position-to-term
| Aspect | Term-to-term rule | Position-to-term rule |
|---|---|---|
| What it tells you | How to get the next term from the previous one | Direct formula for the $n^{\text{th}}$ term |
| Best when | You progress step-by-step | You need a specific term far along the sequence |
| Typical form | Add/subtract/multiply/divide a fixed value | $an + b$ for linear sequences |
Worked Examples
- Given sequence $4,\ 7,\ 10,\ 13,\ \dots$ find both rules.
Common difference $d = 3$. Term-to-term: add $3$. Position-to-term: observe $\text{Term}_1 = 4$, so
$$\text{Term}_n = 4 + (n-1)3$$ which simplifies to
$$\text{Term}_n = 3n + 1$$
- Given $18,\ 16,\ 14,\ 12,\ 10,\ \dots$ find $\text{Term}_{21}$.
$$\text{Term}_n = -2n + 20$$ So
$$\text{Term}_{21} = -2(21) + 20 = -22$$
Real-world applications
- Finance: regular deposits or withdrawals form arithmetic sequences for balances when interest is ignored.
- Scheduling: repeating shifts or timetables with constant time gaps.
- Computer science: iterating counters or stepping indices in loops.
Quick tips
- If the difference between terms is constant, use arithmetic (term-to-term and position-to-term are linear).
- To find a posi
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Sequences Basics
Klíčové pojmy: A sequence is an ordered list of numbers, Term-to-term rules give the operation to get the next term, Arithmetic sequences have constant difference $d$, Position-to-term gives $\text{Term}_n$ as a function of $n$, For arithmetic sequences $\text{Term}_n = a_1 + (n-1)d$, To find linear rule solve $\text{Term}_n = an + b$ using known terms, Check formulas by substituting $n=1$ and another $n$, Use term-to-term for stepwise work and position-to-term for distant terms, Function machines map input $n$ to output term with algebra, Real-world use: finance, scheduling, and programming