A sequence is a function from the whole numbers to the real numbers. Its inputs are called the term numbers and its outputs are called the terms of the sequence.

Define arithmetic sequence .

An arithmetic sequence is a sequence in which successive terms have a common difference . The same value is added to each term to get the next, so the common difference is a constant rate of change.

What can you say about the points on the graph of an arithmetic sequence?

They lie along a straight line , because the common difference acts as a constant slope. The graph is still only those separate points, not the line itself.

Define geometric sequence .

A geometric sequence is a sequence in which successive terms have a common ratio . Each term is found by multiplying the previous term by the same constant, so the change is a constant proportional change.

True or False? If the differences between successive outputs are not constant, the data must be exponential.

False. Failing the difference test only rules a linear model out, and it establishes nothing in its place. You still have to check whether the ratios of successive outputs are constant, and they may well not be.

How many data points do you need to determine a linear or an exponential function completely?

Two, provided you already know which of the two types you are dealing with. Two points give the slope of a linear function, or the base of an exponential function, and either is then enough to write the function out in full.

Why does an exponential function in general form have no local maximum or minimum?

Because it is always increasing or always decreasing, so it never turns around. Its outputs change proportionally over equal-length input intervals, and that rules out any change of direction.

True or False? The graph of an exponential function in general form can change from concave up to concave down.

False. The graph is always concave up or always concave down, and its concavity never changes. That is why the graph of an exponential function has no point of inflection.

AP®MathsCollege BoardPrecalculusFlashcardsExponential & Logarithmic FunctionsSequences & Exponential Functions

Sequences & Exponential Functions (College Board AP® Precalculus): Flashcards

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FrontSequence Basics
Define sequence.

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Cards in this collection (56)

Define sequence.
A sequence is a function from the whole numbers to the real numbers.
Its inputs are called the term numbers and its outputs are called the terms of the sequence.
Fill in the two gaps.
A sequence is a function whose domain is the $______$ numbers, and that domain is a $______$ set of isolated values.
The completed sentence is:
A sequence is a function whose domain is the whole numbers, and that domain is a discrete set of isolated values.
Because of that, the graph of a sequence is a set of separate points rather than a joined-up curve.
The sequence $a_{n}$ is given by $a_{n} = 2 n + 3$ . Find $a_{0}$ and $a_{4}$ .
$a_{0} = 3$ and $a_{4} = 11$ .
Substitute the term number in place of $n$ , giving $2 (0) + 3$ and $2 (4) + 3$ .
True or False?
The terms of a sequence must always start with $a_{1}$ .
False.
A sequence is a function from the whole numbers, and the whole numbers begin at $0$ , so the terms are usually written $a_{0}, a_{1}, a_{2}, \dots$
A sequence is sometimes written starting from $a_{1}$ instead, so always check which term the labelling starts at.
Why does a sequence have exactly one term for each term number?
Because a sequence is a function, and a function assigns exactly one output to each input.
So $a_{5}$ names a single value, never two.
What does it mean to give a formula for the $n$ th term of a sequence?
It means a rule written in terms of $n$ that produces any term you want by substitution.
For $a_{n} = 2 n + 3$ , substituting $n = 0, 1, 2, \dots$ generates the terms $3, 5, 7, \dots$
Define arithmetic sequence.
An arithmetic sequence is a sequence in which successive terms have a common difference.
The same value is added to each term to get the next, so the common difference is a constant rate of change.
An arithmetic sequence has initial value $a_{0}$ and common difference $d$ . Fill in the two gaps in its general term.
$a_{n} =______+______n$
The completed formula is:
$a_{n} = a_{0} + d n$
The initial value $a_{0}$ is the term at $n = 0$ , and $d$ is added once for every step along the sequence.
How do you find the common difference of an arithmetic sequence from a list of its terms?
Subtract any term from the term after it, so $d = a_{n + 1} - a_{n}$ .
Check more than one pair, because if the differences between consecutive terms are not all equal then the sequence is not arithmetic.
An arithmetic sequence has $a_{4} = 20$ and $d = - 2$ . Write a formula for $a_{n}$ .
$a_{n} = 28 - 2 n$ .
Starting from the known term $20$ and stepping $n - 4$ times gives $20 - 2 (n - 4)$ , which expands to $28 - 2 n$ .
Check it against the term you were given: $a_{4} = 28 - 8 = 20$ .
True or False?
A sequence whose common difference is $0$ is not arithmetic.
False.
A common difference of $0$ is perfectly allowed, and it gives a sequence in which every term is the same.
The common difference may be positive, negative or zero.
What can you say about the points on the graph of an arithmetic sequence?
They lie along a straight line, because the common difference acts as a constant slope.
The graph is still only those separate points, not the line itself.
The general term of an arithmetic sequence can be written as $a_{n} = a_{0} + d n$ or as $a_{n} = a_{k} + d (n - k)$ . What single idea do both express?
Start from a term you already know, then add the common difference once for every step from that term to the $n$ th term.
The first form starts from the initial value, and the second starts from any known term, which is what you need when $a_{0}$ has not been given.
An arithmetic sequence is $a_{n} = a_{0} + d n$ and a linear function is $f (x) = b + m x$ . Which part of one corresponds to which part of the other?
$a_{0}$ and $b$ are both the initial value, the output when the input is $0$ .
$d$ and $m$ are both the constant rate of change, appearing as the common difference of the sequence and as the slope of the line.
Why do an arithmetic sequence and a linear function behave in the same way?
Because in both, increasing the input by $1$ adds the same constant amount to the output.
For $a_{n} = 5 + 3 n$ and $f (x) = 5 + 3 x$ , every unit step in the input adds $3$ .
The terms of $a_{n} = 3 + 2 n$ are plotted on the same axes as the graph of $f (x) = 3 + 2 x$ . What do you see?
The plotted points lie exactly on the line, at the whole-number inputs $0, 1, 2, 3, \dots$
The terms of the sequence are the values of the corresponding linear function, read off at those inputs.
True or False?
An arithmetic sequence and the linear function with the same formula have the same domain.
False.
The sequence is defined only for the whole numbers $0, 1, 2, 3, \dots$ , while the linear function is defined for every real number.
That is why one graph is a set of separate points and the other is an unbroken straight line.
An arithmetic sequence based on a known term is $a_{n} = a_{k} + d (n - k)$ . Fill in the two gaps in the matching point-slope form of a linear function.
$f (x) = y_{i} +______(x -______)$
The completed formula is:
$f (x) = y_{i} + m (x - x_{i})$
Here $(x_{i}, y_{i})$ is a known point and $m$ is the slope, so two data points are enough: find $m$ from them, then substitute one of the points.
A tank holds $150$ gallons at $t = 0$ , then $210$ , $270$ and $330$ gallons at $t = 1, 2, 3$ hours. Write a linear model $V (t)$ and find $V (2.5)$ .
$V (t) = 150 + 60 t$ , and $V (2.5) = 300$ gallons.
The initial value is $150$ gallons and the volume rises by $60$ gallons each hour, so $60$ is the constant rate of change.
Define geometric sequence.
A geometric sequence is a sequence in which successive terms have a common ratio.
Each term is found by multiplying the previous term by the same constant, so the change is a constant proportional change.
How do you find the common ratio of a geometric sequence from a list of its terms?
Divide any term by the term before it, so $r = \frac{g_{n + 1}}{g_{n}}$ .
Check more than one pair, because if the ratios between consecutive terms are not all equal then the sequence is not geometric.
The geometric sequence $3, 6, 12, 24, \dots$ has the general term below. Fill in the two gaps, the second of which is the base of the power.
$g_{n} =______\cdot______^{n}$
The completed formula is:
$g_{n} = 3 \cdot 2^{n}$
The initial value is the term at $n = 0$ , which is $3$ , and the common ratio is $\frac{6}{3} = 2$ .
A geometric sequence has $g_{2} = 48$ and $r = \frac{1}{2}$ . Write a formula for $g_{n}$ , and find $g_{0}$ .
$g_{n} = 48 \cdot {(\frac{1}{2})}^{(n - 2)}$ , and $g_{0} = 192$ .
Starting from the known term $48$ at position $2$ , multiply by the common ratio once for every step from there to position $n$ .
Reaching $g_{0}$ means two steps backwards, so $48 \cdot {(\frac{1}{2})}^{- 2} = 48 \cdot 4 = 192$ .
True or False?
The common ratio of a geometric sequence can be negative.
True.
A negative common ratio makes the terms alternate in sign, as in $3, - 6, 12, - 24, \dots$ where $r = - 2$ .
The one value a common ratio cannot take is zero.
How does the step size of an increasing geometric sequence compare with that of an increasing arithmetic sequence?
An increasing arithmetic sequence goes up by the same amount at every step, while an increasing geometric sequence goes up by a larger amount at each successive step.
That happens because the common ratio is applied to an ever-larger value, so $2, 6, 18, 54$ rises by $4$ , then $12$ , then $36$ .
For a geometric sequence with $g_{0} > 0$ , what do the plotted points look like when $r > 1$ , and when $0 < r < 1$ ?
When $r > 1$ the points curve upward, getting steeper as $n$ increases.
When $0 < r < 1$ the points curve downward, getting closer and closer to zero.
A geometric sequence is $g_{n} = g_{0} r^{n}$ and an exponential function is $f (x) = a b^{x}$ . Which part of one corresponds to which part of the other?
$g_{0}$ and $a$ are both the initial value, the output when the input is $0$ .
$r$ and $b$ are both the constant ratio, appearing as the common ratio of the sequence and as the base of the exponential function.
Why do a geometric sequence and an exponential function behave in the same way?
Because in both, increasing the input by $1$ multiplies the output by the same constant.
For $g_{n} = 5 \cdot 2^{n}$ and $f (x) = 5 \cdot 2^{x}$ , every unit step in the input doubles the value.
A geometric sequence based on a known term is $g_{n} = g_{k} r^{(n - k)}$ . Fill in the two gaps in the matching known-point form of an exponential function.
$f (x) =______b^{(x -______)}$
The completed formula is:
$f (x) = y_{i} b^{(x - x_{i})}$
Here $(x_{i}, y_{i})$ is a known point on the graph and $b$ is the base, matching the known term $g_{k}$ and the common ratio $r$ .
You know a relationship is exponential and you have two data points. How do you write the function?
Find the base $b$ from the ratio of the two output values, allowing for how far apart the two inputs are.
Then take either point as $(x_{i}, y_{i})$ and substitute it into the known-point form.
A colony holds $8$ , $12$ , $18$ and $27$ thousand bacteria at $t = 0, 1, 2, 3$ hours. Write an exponential model $B (t)$ and find $B (2.5)$ .
$B (t) = 8 \cdot 1 . 5^{t}$ , and $B (2.5) \approx 22.045$ thousand bacteria.
The initial value is $8$ , and each hour multiplies the count by $\frac{12}{8} = 1.5$ , so $1.5$ is the base.
True or False?
Every geometric sequence with a positive common ratio can be extended to an exponential function agreeing with it at every term.
True.
Take the initial value as $a$ and the common ratio as $b$ , and $f (x) = a b^{x}$ reproduces every term at the whole-number inputs.
The restriction matters: a sequence with a negative common ratio has no such partner, because the base of an exponential function must be positive.
Two functions have outputs $4, 10, 16, 22$ and $4, 8, 16, 32$ at $x = 0, 1, 2, 3$ . How do you decide which is linear and which is exponential?
Over equal-length input intervals, constant differences mean linear and constant ratios mean exponential.
The first set has differences $6, 6, 6$ so it is linear, and the second has ratios $2, 2, 2$ so it is exponential.
What do a linear function and an exponential function have in common?
Both are built from an initial value together with one constant that controls how the output changes.
In $f (x) = b + m x$ that constant is the slope $m$ , and in $f (x) = a b^{x}$ it is the base $b$ .
Fill in the two gaps.
A linear function is built on repeated $______$ of a fixed amount, and an exponential function is built on repeated $______$ by a fixed factor.
The completed sentence is:
A linear function is built on repeated addition of a fixed amount, and an exponential function is built on repeated multiplication by a fixed factor.
That difference in operation is the only structural difference between the two families.
True or False?
If the differences between successive outputs are not constant, the data must be exponential.
False.
Failing the difference test only rules a linear model out, and it establishes nothing in its place.
You still have to check whether the ratios of successive outputs are constant, and they may well not be.
How many data points do you need to determine a linear or an exponential function completely?
Two, provided you already know which of the two types you are dealing with.
Two points give the slope of a linear function, or the base of an exponential function, and either is then enough to write the function out in full.
Why can the base of an exponential function not be $1$ ?
Because $1$ raised to any power is $1$ , so every output would equal the initial value.
That is a constant function rather than growth or decay, which is why the base must satisfy $b > 0$ and $b \neq 1$ .
An exponential model has value $128, 000$ at $t = 4$ and $18, 000$ at $t = 10$ . How can you find its value at $t = 7$ without first finding $a$ and $b$ ?
The value at $t = 7$ is $48, 000$ .
Outputs over equal-length intervals form a geometric sequence, so $128, 000 \cdot r^{6} = 18, 000$ , which gives $r^{6} = \frac{9}{64}$ and so $r^{3} = \frac{3}{8}$ .
Three equal steps run from $t = 4$ to $t = 7$ , so the value there is $128, 000 \cdot \frac{3}{8}$ .
Fill in the two gaps.
An exponential function has general form $f (x) = a b^{x}$ , in which $a$ is the $______$ value and $b$ is the $______$ of the function.
The completed sentence is:
An exponential function has general form $f (x) = a b^{x}$ , in which $a$ is the initial value and $b$ is the base of the function.
The base must be positive.
In $f (x) = a b^{x}$ , why is $f (0) = a$ , and why can $a$ not be $0$ ?
$f (0) = a b^{0} = a$ , because a positive base raised to the power $0$ is $1$ .
If $a$ were $0$ then every output would be $0$ , giving the constant zero function rather than an exponential one.
Fill in the two gaps.
For $f (x) = a b^{x}$ with $a > 0$ , a base $b > 1$ gives exponential $______$ as $x$ increases, and a base $0 < b < 1$ gives exponential $______$ instead.
The completed sentence is:
For $f (x) = a b^{x}$ with $a > 0$ , a base $b > 1$ gives exponential growth as $x$ increases, and a base $0 < b < 1$ gives exponential decay instead.
With $b > 1$ each unit step multiplies by more than $1$ , and with $0 < b < 1$ each step multiplies by less than $1$ .
Why does an exponential function in general form have no local maximum or minimum?
Because it is always increasing or always decreasing, so it never turns around.
Its outputs change proportionally over equal-length input intervals, and that rules out any change of direction.
True or False?
The graph of an exponential function in general form can change from concave up to concave down.
False.
The graph is always concave up or always concave down, and its concavity never changes.
That is why the graph of an exponential function has no point of inflection.
The function $f$ is given by $f (x) = a b^{x}$ with $a > 0$ and $b > 1$ . Write its end behavior using limit notation.
The two limits are:
$\lim_{x \to \infty} a b^{x} = \infty$
$\lim_{x \to - \infty} a b^{x} = 0$
So the output increases without bound as $x$ increases, and gets arbitrarily close to zero as $x$ decreases without bound.
How does the end behavior of $a b^{x}$ change if $a$ is negative instead of positive?
The limit that was $\infty$ becomes $- \infty$ , and the limit that was $0$ stays $0$ .
Multiplying by a negative $a$ flips the sign of an output that grows without bound, but any multiple of something approaching zero still approaches zero.
The outputs $7, 9, 13, 21$ at $x = 1, 2, 3, 4$ do not have constant ratios. How can they still be modeled by an exponential function?
Subtracting $5$ from every output leaves $2, 4, 8, 16$ , whose ratios are all $2$ .
So the data fits $f (x) = 2^{x} + 5$ , an exponential function with a constant added to it.
When ratios are not constant, test whether adding or subtracting a constant makes them constant.
Define the natural base $e$ .
The natural base $e$ is an irrational constant, approximately $2.718$ , whose decimal expansion never ends and never repeats.
It is the base most often used in exponential models of growth and decay.
Where does $e$ fit into the general form $a b^{x}$ , and does $f (x) = e^{x}$ grow or decay?
It is the base, so an exponential function with base $e$ is written $f (x) = a e^{x}$ .
Since $e \approx 2.718$ is greater than $1$ , the function $f (x) = e^{x}$ shows exponential growth.
Fill in the two gaps.
When you multiply two powers with the same base you $______$ the exponents, and when you raise a power to another power you $______$ them.
The completed sentence is:
When you multiply two powers with the same base you add the exponents, and when you raise a power to another power you multiply them.
These are the product property $b^{m} \cdot b^{n} = b^{(m + n)}$ and the power property ${(b^{m})}^{n} = b^{m n}$ .
Why is every horizontal translation of $b^{x}$ the same as a vertical dilation?
By the product property $b^{(x + k)} = b^{k} \cdot b^{x}$ , and $b^{k}$ is simply a constant.
Multiplying by a constant is a vertical dilation, so $3^{(x + 2)} = 9 \cdot 3^{x}$ : shifting left by $2$ is the same as stretching by a factor of $9$ .
True or False?
Dilating the graph of $b^{x}$ horizontally changes the base of the function.
True.
By the power property $b^{(c x)} = {(b^{c})}^{x}$ , and $b^{c}$ is a constant, so the result is an exponential function with a new base.
For instance $3^{(2 x)} = 9^{x}$ , so compressing $3^{x}$ horizontally by a factor of $\frac{1}{2}$ turns the base into $9$ .
What does a negative exponent mean, and how does it turn a growth form into a decay form?
A negative exponent gives the reciprocal of the positive power, so $b^{- n} = \frac{1}{b^{n}}$ .
Applied to a function, $2^{- x} = {(\frac{1}{2})}^{x}$ , so a base greater than $1$ with a negative exponent is the same as a base between $0$ and $1$ .
What does $b^{\frac{1}{k}}$ mean, and what is the value of $8^{\frac{2}{3}}$ ?
$b^{\frac{1}{k}}$ is the $k$ th root of $b$ , when that root exists, so $b^{\frac{1}{2}} = \sqrt{b}$ and $b^{\frac{1}{3}} = \sqrt[3]{b}$ .
Combining that with the power property, $8^{\frac{2}{3}} = {(\sqrt[3]{8})}^{2} = 2^{2} = 4$ .
How do you rewrite $4^{x}$ as a power of $2$ ?
Write the original base as a power of the new one, $4 = 2^{2}$ , then apply the power property.
That gives $4^{x} = {(2^{2})}^{x} = 2^{2 x}$ .
Rewrite $4 \cdot 9^{(x + 1)}$ in the form $c \cdot 3^{m x}$ .
$4 \cdot 9^{(x + 1)} = 36 \cdot 3^{(2 x)}$ .
First split the exponent, $9^{(x + 1)} = 9 \cdot 9^{x}$ , so the whole expression becomes $36 \cdot 9^{x}$ .
Then change the base using $9 = 3^{2}$ , which gives $9^{x} = {(3^{2})}^{x} = 3^{(2 x)}$ .

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