Summary of Linear Sequences and Basic Functions
Linear Sequences & Basic Functions: N Term Rules Explained
Introduction
Sequences and functions are two core ideas in algebra that link numbers and patterns. A sequence is an ordered list of numbers following a rule. A function is a relationship that assigns each input a single output. Understanding how to find rules for sequences and write functions helps solve many real-world and mathematical problems.
Sequences: basics and the nth term
What is a sequence?
A sequence is an ordered list of numbers that often follows a clear rule.
- Example: $3,;5,;7,;9,\dots$ is a sequence where each term increases by $2$.
- We usually label terms as $T_1,;T_2,;T_3,\dots$ or use $a_n$ for the $n^{\text{th}}$ term.
The nth-term (general term)
The nth term of a sequence is an expression that gives the $n^{\text{th}}$ term for any integer $n\ge 1$.
- For the sequence $3,;5,;7,;9,\dots$ the difference between terms is $2$, so the rule is $a_n = 2n+1$. This gives $a_1=3$, $a_2=5$, etc.
Example: Use the rule $a_n = 2n+1$ to find the number of matches needed for the $20^{\text{th}}$ pattern. $$a_{20} = 2\times 20 + 1$$ $$a_{20} = 41$$
Checking if a number appears in a sequence
- To check whether a number $M$ is a term of $a_n$, solve $a_n = M$ for integer $n$.
Example: Is $85$ a term in the sequence $a_n = 5n+8$? Solve: $$5n+8 = 85$$ $$5n = 77$$ $$n = \frac{77}{5} = 15.4$$ Since $n$ is not a whole number, $85$ is not a term of that sequence.
Arithmetic sequences
- An arithmetic sequence has a constant difference $d$ between consecutive terms.
- General form: $$a_n = a_1 + (n-1)d$$
Example: $26,;21,;16,;11,;6,\dots$ has common difference $d=-5$. If $a_1=26$, then $$a_n = 26 + (n-1)(-5)$$ which expands to $$a_n = -5n + 31$$
Other patterns and care with data
- Not every list with varying jumps is a single simple arithmetic sequence. Check differences carefully.
- If differences vary or alternate, you may need to look for other rules or piecewise definitions.
Functions: input-output relationships
What is a function?
A function is a rule that assigns each input exactly one output.
- Inputs are often called $x$ and outputs $y$ or $f(x)$.
- Example linear function: $y = 2x + 1$ means each input $x$ maps to $2x+1$.
Tables and functions
- You can represent a function by a table of input-output pairs.
Example table for $y = 2x + 1$ with $x=0,1,2,3,4$:
| $x$ | $y=2x+1$ |
|---|---|
| 0 | 1 |
| 1 | 3 |
| 2 | 5 |
| 3 | 7 |
| 4 | 9 |
- Read the table as: input $0$ gives output $1$, input $1$ gives output $3$, and so on.
Finding a function from data
- Look for patterns in outputs as $x$ increases. If outputs change by a constant amount, the function is linear.
- Example: If outputs increase by $2$ when $x$ increases by $1$, the rule is $y=2x+$ constant. Determine the constant by substituting one pair.
Writing and using linear functions
- General linear function: $$y = mx + c$$ where $m$ is the gradient (slope) and $c$ is the y-intercept.
Example: A function that doubles $x$ then adds $1$ is $$y = 2x + 1$$
- If $x=5$, $$y = 2\times 5 + 1 = 11$$
Visual and real-world applications
- Sequences model steps in a process: number of matches in growing patterns, seating in rows, or levels in a pyramid.
- Functions model relationships: cost per item ($y=px+f$), conversion formulas, or simple physics relations like distance under constant speed ($s = vt$).
Comparing sequences and functions
| Aspect | Sequence | Function |
|---|---|---|
| Main idea | Ordered list of terms | Mapping from inputs to outputs |
| Typical nota |
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Sequences and Functions
Klíčové pojmy: A sequence is an ordered list with a rule for each term, The nth term $a_n$ gives the $n^{\text{th}}$ term for $n\ge 1$, Arithmetic sequence formula: $a_n = a_1 + (n-1)d$, Find difference between consecutive terms to identify arithmetic sequences, To test membership solve $a_n = M$ and check $n$ is a whole number, Linear function general form: $y = mx + c$, From table find slope by change in $y$ over change in $x$ and solve for intercept, Use sequences and functions to model real-world patterns and relations