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Arithmetic and geometric sequences as functions of position, with explicit and recursive rules, and the shifts, stretches and reflections of a graph.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you treat a sequence as a function whose input is the position in the list, so an arithmetic sequence turns out to be a linear function and a geometric one an exponential function — and the awkward $n - 1$ in both rules stops being arbitrary. You then learn the two rules that govern every transformation of a graph: outside the function is vertical and reads the way it looks, inside the brackets is horizontal and does the opposite.
You already continue a pattern like $5, 9, 13, \ldots$ by spotting that each step adds $4$, and you already write $2^7$ for repeated multiplication. A sequence is a function whose input is the position, $1, 2, 3, \ldots$, so everything you know about $f(n)$ applies here unchanged.
Sequence: an ordered list of numbers. The $n$th one is written $a(n)$ or $a_n$.
Arithmetic sequence: each term is the previous one plus a fixed common difference $d$.
Geometric sequence: each term is the previous one times a fixed common ratio $r$.
Explicit rule: gives $a(n)$ straight from $n$. Recursive rule: gives $a(n)$ from $a(n-1)$, plus a starting value.
An arithmetic sequence adds the same amount each step. Its explicit rule is $a(n) = a(1) + (n - 1)d$, which multiplies out to $dn$ plus a constant — so an arithmetic sequence is a linear function of $n$, and $d$ is its slope. A geometric sequence multiplies by the same factor each step, so $a(n) = a(1) \cdot r^{\,n-1}$, an exponential function of $n$. Both exponents and multipliers carry $n - 1$ rather than $n$, because there are $n - 1$ steps between the first term and the $n$th. A recursive rule says the same thing differently: give the starting value and say what one step does — $a(1) = 2$, $a(n) = 3a(n-1)$. To tell the two kinds apart, subtract consecutive terms; if the differences are not constant, divide instead.
Another way: diagram
Two rows of dots: the top row labelled $5, 9, 13, 17$ with $+4$ arcs between them, the bottom labelled $3, 6, 12, 24$ with $\times 2$ arcs, and each row's step count marked $1, 2, 3$.
Another way: story
Count the gaps, not the terms. Fence posts and fence panels: five posts have four panels between them, and the $n$th term is reached after $n - 1$ steps.
"The $20$th term of $5, 9, 13, \ldots$ is $5 + 20 \times 4$." It takes only $19$ steps to reach the $20$th term: $5 + 19 \times 4 = 81$. The off-by-one is the commonest error in the whole topic.
"The rule for $7, 10, 13, \ldots$ is $3n$." At $n = 1$ that gives $3$, not $7$. The rule is $3n + 4$: the common difference times $n$, plus whatever makes the first term come out right.
"A recursive rule is enough on its own." $a(n) = 3a(n-1)$ describes $2, 6, 18, \ldots$ and also $5, 15, 45, \ldots$. Without $a(1)$ it names no sequence at all.
The common difference is $9 - 5 = 4$.
Subtract consecutive terms.
Reaching the $20$th term takes $19$ steps, not $20$.
Count the gaps.
$5 + 19 \times 4 = 5 + 76 = 81$.
The ratio is $6 \div 3 = 2$, so this is geometric.
Differences are not constant, so divide.
The $8$th term is $3 \times 2^7$, with $7$ multiplications.
One less than the term number again.
$2^7 = 128$, so the term is $3 \times 128 = 384$.
The common difference is $3$, so the rule contains $3n$.
At $n = 1$, $3n$ gives $3$ but the term is $7$, so the rule is $3n + 4$.
An arithmetic sequence starts $5$, $11$, $17$, $\ldots$ What is its $21$th term?
answer
A geometric sequence starts $3$, $6$, $12$, $\ldots$ What is its $6$th term?
Answer:
Write the $n$th term of $11$, $19$, $27$, $35$, $\ldots$ as an expression in $n$.
Answer:
Write a recursive rule for $5$, $20$, $80$, $320$, $\ldots$ Fill in both blanks.
$a(1) = $ a $,\quad a(n) = $ r $\times a(n - 1)$
You already evaluate $f(x)$ by substituting, and you already know that adding to $y$ moves a line up. That is the whole of this topic, applied to any graph at all rather than only to lines — and the one genuinely surprising part is what happens when the change is made to the input instead of the output.
Transformation: a change to a function that moves, stretches or flips its graph.
Translation: a slide, with no change of shape — up, down, left or right.
Stretch / compression: the graph is scaled away from or towards an axis.
Reflection: a flip across an axis. $-f(x)$ flips vertically; $f(-x)$ flips horizontally.
Parent function: the plain version — $f(x) = x^2$, say — that the transformed one is compared with.
Every transformation is one of two kinds, and which one it is depends on where the change is written. A change outside $f$ acts on the output and moves the graph vertically in the obvious direction: $f(x) + 3$ goes up $3$, $2f(x)$ stretches vertically by $2$, and $-f(x)$ reflects in the $x$-axis. A change inside the brackets acts on the input and moves the graph horizontally in the opposite direction to what the sign suggests: $f(x - 2)$ goes right $2$, $f(2x)$ compresses horizontally by $2$, and $f(-x)$ reflects in the $y$-axis. The reason is the same in every case: to get the output the old graph had, the new input must be adjusted to undo whatever the brackets do to it.
Another way: diagram
The curve $y = x^2$ drawn in grey, with $y = x^2 + 3$ above it in one colour and $y = (x - 2)^2$ to its right in another, each labelled with its rule and the direction it moved.
Another way: story
Think of the brackets as a doorway the input passes through. Anything written in the doorway happens to the input before the function sees it, so the graph has to move the other way to compensate; anything written outside happens afterwards, to the answer.
"$f(x - 2)$ shifts the graph left, because of the minus." It shifts it right. To get the output $f$ used to give at $0$, the new input must be $2$: everything happens two later.
"$-f(x)$ and $f(-x)$ are the same." The first negates the output and flips the graph over the $x$-axis; the second negates the input and flips it over the $y$-axis.
"$f(2x)$ stretches the graph out." It squashes it. The input reaches any given value twice as soon, so the graph is compressed horizontally by a factor of $2$.
The $+3$ is outside the function, so it acts on the output.
Outside means vertical.
Every output is $3$ larger, so every point of the graph is $3$ higher: the graph shifts up $3$.
The change is inside the brackets, so it acts on the input.
Inside means horizontal.
To get what $f$ gave at $0$, the new input must satisfy $x - 2 = 0$, so $x = 2$.
Ask which new input reproduces an old output.
Everything happens $2$ later, so the graph shifts right $2$.
The change is inside the brackets, so the movement is horizontal.
What $f$ gave at $0$ now happens when $x + 3 = 0$, that is at $x = -3$: the graph shifts left $3$.
How does the graph of $f(x) + 9$ compare with the graph of $f(x)$?
How does the graph of $f(x + 1)$ compare with the graph of $f(x)$?
If $f(1) = 5$, how does the graph of $-f(x)$ compare with the graph of $f(x)$?
Match each rule to what it does to the graph of $f$.
| vertical stretch by $6$ | horizontal compression by $6$ | shift down $2$ | reflection in the $y$-axis | |
|---|---|---|---|---|
| $6f(x)$ | ||||
| $f(6x)$ | ||||
| $f(x) - 2$ | ||||
| $f(-x)$ |
Last year a café's daily takings on day $d$ of the month were $S(d)$. This year the same pattern happens, unchanged in size, but everything falls exactly $4$ days later in the month. Which rule gives this year's takings on day $d$?
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
Is $3$, $12$, $48$, $192$, $\ldots$ arithmetic or geometric?
Suppose $f(4) = -3$. What is $4f(4) - 5$?
Answer:
You can find the terms and rules of arithmetic and geometric sequences and describe how a transformation moves a graph. Explain why the $20$th term of $5, 9, 13, \ldots$ uses $19$ steps, and why $f(x - 2)$ shifts the graph right rather than left.
8. Your turn: write the $n$th term of $7, 10, 13, 16, \ldots$, step 2
19. Your turn: how does $f(x + 3)$ compare with $f(x)$?, step 2