1.5.1 Sequences | SigmaField

What a sequence is

A sequence is a list of numbers in a fixed order. Each number is a term, and $a_n$ is the term in position $n$ , as with the subscripts of lesson 0.2.1. Positions are whole numbers, usually starting at 1.

For $a_n = 2n + 1$ the terms are 3, 5, 7, 9, … The fourth term is $a_4 = 2 \times 4 + 1 = 9$ .

The terms carry the units of whatever they measure, such as dollars or metres. The position $n$ is a count and has no units.

A sequence is a function whose inputs are $1, 2, 3, \ldots$ (lesson 1.1.1). So its graph is a row of separate dots: nothing sits between positions 2 and 3.

Two ways to define a sequence

An explicit definition gives a formula for $a_n$ in terms of $n$ . You can jump straight to any term: for $a_n = 2n + 1$ , the 100th term is 201.

A recursive definition gives a first term and a rule that builds each term from the one before. The rule $a_1 = 3$ , $a_{n+1} = a_n + 2$ makes the same 3, 5, 7, 9, … To reach the 100th term this way, you must build all 99 terms before it.

Rule: Explicit and recursive

\begin{gathered} \text{explicit: } a_n = 2n + 1 \\ \text{recursive: } a_1 = 3, \ \ a_{n+1} = a_n + 2 \end{gathered}

Read $a_{n+1}$ as "the next term". A recursive rule needs a starting term, or it defines nothing.

A recursive rule can hide a formula. With $a_1 = 1$ and $a_{n+1} = 2a_n + 1$ , the terms are 1, 3, 7, 15, 31, 63. Each is one less than a power of 2, so $a_n = 2^n - 1$ .

2a_n + 1

print([2 * n + 1 for n in range(1, 7)]) a = 3 # a_1 terms = [a] for step in range(5): a = a + 2 # a_(n+1) = a_n + 2 terms.append(a) print(terms) a = 1 terms = [a] for step in range(5): a = 2 * a + 1 terms.append(a) print(terms, [2**n - 1 for n in range(1, 7)])

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Arithmetic sequences

Start at 3 and add 4 each time: 3, 7, 11, 15, 19. Adding the same common difference $d$ at every step makes an arithmetic sequence. The difference can be positive, negative or zero.

Rule: Arithmetic sequence

\begin{gathered} a_n = a_1 + (n - 1)\,d \\ a_{n+1} = a_n + d \end{gathered}

To reach term $n$ you add $d$ exactly $n - 1$ times. The first term $a_1$ needs no steps at all.

For 3, 7, 11, …, the 20th term is $3 + 19 \times 4 = 79$ . To find which term equals 99, solve $3 + 4(n - 1) = 99$ , as in lesson 0.3.2. That gives $n - 1 = 24$ , so 99 is the 25th term.

A negative difference makes the terms fall. In 50, 46, 42, … the difference is $-4$ . The first negative term needs $50 - 4(n - 1) < 0$ , so $n > 13.5$ : it is the 14th term, $-2$ .

On a graph, the dots of an arithmetic sequence lie on a straight line with slope $d$ (lesson 1.2.1).

Geometric sequences

Start at 5 and double each time: 5, 10, 20, 40, 80. Multiplying by the same common ratio $r$ at every step makes a geometric sequence. The ratio is never 0.

Rule: Geometric sequence

\begin{gathered} a_n = a_1\, r^{\,n - 1} \\ a_{n+1} = r\, a_n \end{gathered}

To reach term $n$ you multiply by $r$ exactly $n - 1$ times. The 10th term of 5, 10, 20, … is $5 \times 2^9 = 2560$ .

Growth by a fixed percentage is geometric (lesson 1.4.2). Growing by 10% a step means $r = 1.1$ : 100, 110, 121, 133.1. Shrinking by 20% a step means $r = 0.8$ .

A negative ratio flips the sign at every step. With $a_1 = 8$ and $r = -\tfrac{1}{2}$ the terms are $8, -4, 2, -1, 0.5, \ldots$

Telling them apart

Subtract each term from the next. If the differences are all equal, the sequence is arithmetic. If instead the ratios $a_{n+1} / a_n$ are all equal, it is geometric.

sequence	differences	ratios	type
2, 6, 10, 14	4, 4, 4	3, 1.67, 1.4	arithmetic
2, 6, 18, 54	4, 12, 36	3, 3, 3	geometric
1, 4, 9, 16	3, 5, 7	4, 2.25, 1.78	neither

Try it zip(s, s[1:]) pairs each term with the next one. Then a difference is b - a and a ratio is b / a . The cell rounds the ratios to 4 places; the table rounds them to 2.

def differences(s): return [b - a for a, b in zip(s, s[1:])] def ratios(s): return [round(b / a, 4) for a, b in zip(s, s[1:])] for s in [[2, 6, 10, 14], [2, 6, 18, 54], [1, 4, 9, 16]]: print(s, differences(s), ratios(s)) print(3 + (20 - 1) * 4, 5 * 2 ** (10 - 1)) print([8 * (-0.5) ** (n - 1) for n in range(1, 7)])

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Increasing, decreasing and alternating

A sequence is increasing if every term is bigger than the one before: $a_{n+1} > a_n$ . It is decreasing if every term is smaller: $a_{n+1} < a_n$ . It is alternating if the signs switch at every step.

The test is the sign of the step $a_{n+1} - a_n$ . For an arithmetic sequence the step is $d$ . For a geometric sequence with a positive first term:

$r > 1$ : increasing, as in 5, 10, 20, …
$0 < r < 1$ : decreasing, as in 10, 8, 6.4, …
$r = 1$ : constant, every term the same
$r < 0$ : alternating, as in $8, -4, 2, \ldots$

Watch the sign of the first term. With $a_1 = -1$ and $r = 2$ the terms are $-1, -2, -4, -8$ : they grow in size but decrease. Some sequences are none of these: $a_n = (n - 3)^2$ gives 4, 1, 0, 1, 4, falling and then rising.

Four sequences as dots. Two have a limit, 0; the other two have none, one because it grows for ever and one because it never settles.

The limit of a sequence

A sequence has the limit $L$ if its terms get as close to $L$ as you like and then stay that close. You write $a_n \to L$ , read " $a_n$ tends to $L$ ", or $\lim_{n \to \infty} a_n = L$ . A sequence with a limit converges; one without a limit diverges.

Take $a_n = \frac{n + 1}{n}$ . Its terms fall towards 1.

$n$	1	2	5	10	100	1000
$a_n$	2	1.5	1.2	1.1	1.01	1.001
distance from 1	1	0.5	0.2	0.1	0.01	0.001

Algebra explains the table: $\frac{n + 1}{n} = 1 + \frac{1}{n}$ , so the distance from 1 is exactly $\frac{1}{n}$ . It is less than 0.1 once $n > 10$ , and less than 0.01 once $n > 100$ . However small the tolerance, from some term on every term is that close.

The band reaches 0.1 either side of the limit 1. The 10th term sits on its edge; from the 11th on, every term is inside, and none ever leaves.

Rule: Limit of a sequence

\begin{gathered} a_n \to L \quad\text{or}\quad \lim_{n \to \infty} a_n = L \end{gathered}

For every tolerance, however small, there is a term after which all the terms lie within that tolerance of $L$ . Proof Gym stage PG4 makes this precise.

The four sequences pictured in the previous section show the remaining cases.

Alternating can converge. In $8, -4, 2, -1, \ldots$ the terms land on both sides of 0, ever closer. The limit is 0.
Bouncing is not converging. $-1, 1, -1, 1, \ldots$ keeps visiting 1 but always leaves again, so it has no limit.
Growing without end is diverging. For $3, 7, 11, \ldots$ there is no $L$ to approach. You write $a_n \to \infty$ .

A limit need not be reached: $1 + \frac{1}{n}$ is never exactly 1. It may be reached, too: the constant sequence 5, 5, 5, … has limit 5.

\frac{1}{n}

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Where $r^n$ goes

Powers of a ratio, $r, r^2, r^3, \ldots$ , form the simplest geometric sequence. What they do depends only on $r$ .

$r$	first terms	behaviour	limit
2	2, 4, 8, 16	increasing	none: $r^n \to \infty$
1	1, 1, 1, 1	constant	1
0.5	0.5, 0.25, 0.125, 0.0625	decreasing	0
$-0.5$	$-0.5$ , 0.25, $-0.125$ , 0.0625	alternating	0
$-1$	$-1$ , 1, $-1$ , 1	alternating	none
$-2$	$-2$ , 4, $-8$ , 16	alternating	none

Rule: Powers of the ratio

\begin{gathered} |r| < 1: \ r^n \to 0 \\ r = 1: \ r^n = 1 \\ r > 1: \ r^n \to \infty \\ r \le -1: \ \text{no limit} \end{gathered}

The size $|r|$ (lesson 0.3.3) decides. Below 1, each step shrinks the terms towards 0; at or above 1, they never shrink. The sign of $r$ only decides whether they alternate.

The same holds for $a_1 r^{n-1}$ : multiplying every term by a fixed $a_1$ does not change where they go. The next two lessons rest on the first line of the rule, and case study 1.5.5 uses it to price payments that never stop.

r^n

for r in [2, 1, 0.5, -0.5, -1, -2]: print(r, [r**n for n in range(1, 6)]) for r in [0.5, 0.9, 0.99]: n = 1 while r**n >= 0.001: n = n + 1 print(r, n)

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Common mistakes

Term $n$ needs $n - 1$ steps, not $n$ . The first term is $a_1$ itself.
Arithmetic adds; geometric multiplies. Check the differences and the ratios before using a formula: 1, 4, 9, 16 is neither.
Increasing is about the terms, not their size. $-1, -2, -4$ has ratio 2 but is decreasing.
A negative ratio alternates. It converges only when $|r| < 1$ : $(-0.5)^n \to 0$ , but $(-2)^n$ has no limit.
Getting close once is not a limit. $-1, 1, -1, \ldots$ hits 1 again and again, but leaves each time. The terms must stay close.
A recursive rule needs its first term. $a_{n+1} = a_n + 2$ alone does not say where to start.

Sequences

0 of 12 correct

1.
If $a_n = 3n - 1$ , find $a_{10}$
2.
If $a_1 = 2$ and $a_{n+1} = 3a_n - 1$ , find $a_4$
3.
Find the 10th term of $7, 10, 13, 16, \ldots$
4.
In $2, 6, 10, \ldots$ , which term equals 50?
5.
Find the first negative term of $50, 46, 42, \ldots$
6.
Find the 5th term of $2, 6, 18, \ldots$
7.
What is the common ratio of $8, -4, 2, -1, \ldots$ ?
8.
A sequence starts at $a_1 = 100$ and grows by 10% a step. Find $a_4$
9.
For $a_n = n^2$ , simplify the step $a_{n+1} - a_n$
10.
Find the limit of $a_n = \frac{2n + 3}{n}$
11.
$a_n = 1 + \frac{1}{n}$ has limit 1. From which term on is it closer than 0.001 to 1?
12.
Find the limit of $5 \cdot (-0.8)^n$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

a_1, a_2, \ldots

Sequences

What a sequence is

Two ways to define a sequence

Rule: Explicit and recursive

Arithmetic sequences

Rule: Arithmetic sequence

Geometric sequences

Rule: Geometric sequence

Telling them apart

Increasing, decreasing and alternating

The limit of a sequence

Rule: Limit of a sequence

Where $r^n$ goes

Rule: Powers of the ratio

Common mistakes

Sequences

Summary card

Free further reading

What a sequence is

Two ways to define a sequence

Rule: Explicit and recursive

Arithmetic sequences

Rule: Arithmetic sequence

Geometric sequences

Rule: Geometric sequence

Telling them apart

Increasing, decreasing and alternating

The limit of a sequence

Rule: Limit of a sequence

Where rnr^nrn goes

Rule: Powers of the ratio

Common mistakes

Summary card

Free further reading

Where $r^n$ goes