Sequences

Ordered lists generated by explicit or recursive rules.

A sequence is an ordered list of terms. Unlike a set, order matters. The sequence 2,4,8 is different from 8,4,2.

Learning objectives

After this lesson you should be able to:

  • interpret sequence notation such as a_n;
  • compute terms from explicit formulas;
  • compute terms from recursive formulas;
  • recognize arithmetic and geometric sequences;
  • write simple closed forms;
  • evaluate small finite sums.

Sequence notation

A sequence is often written as

a_1, a_2, a_3, ...

The subscript tells the position. The value at that position is the term.

TipPosition versus value

In the sequence 5, 8, 11, the third term is 11. The position is 3; the value is 11.

Explicit rules

An explicit rule gives a term directly from its position.

Example:

a_n = 3n + 2

Then

a_1 = 5
a_2 = 8
a_3 = 11

Recursive rules

A recursive rule gives starting term(s) and a way to get later terms from earlier terms.

Example:

a_1 = 5
a_{n+1} = a_n + 3

This generates 5, 8, 11, 14, ....

Arithmetic sequences

An arithmetic sequence adds the same number each time. That number is the common difference.

If the first term is a_1 and the common difference is d, then

a_n = a_1 + (n-1)d

Geometric sequences

A geometric sequence multiplies by the same number each time. That number is the common ratio.

If the first term is a_1 and the common ratio is r, then

a_n = a_1 r^(n-1)

Finite sums

Sigma notation adds terms:

sum from i=1 to 4 of i = 1 + 2 + 3 + 4 = 10

The index variable is temporary. It tells you which terms to add.

Worked example 1: arithmetic sequence

Find the 10th term of 4, 9, 14, ....

The first term is a_1=4. The common difference is d=5.

a_10 = 4 + (10-1)5
     = 4 + 45
     = 49

Worked example 2: geometric sequence

Find the next term of 3, 6, 12, 24, ....

Each term is multiplied by 2, so the next term is

24 * 2 = 48

Worked example 3: recursive rule

Suppose

a_1 = 2
a_{n+1} = 2a_n + 1

Then

a_2 = 2(2)+1 = 5
a_3 = 2(5)+1 = 11
a_4 = 2(11)+1 = 23

Common pitfalls

  • Confusing term number with term value.
  • Assuming every sequence starts at n=1.
  • Using an arithmetic formula for a geometric pattern.
  • Forgetting the starting value in a recursive rule.
  • Treating a sequence like an unordered set.

Practice

  1. Find the next term: 2, 5, 8, 11, ....
  2. Find the next term: 3, 6, 12, 24, ....
  3. Find a_5 for a_n = 2n + 1.
  4. If a_1=1 and a_{n+1}=a_n+4, find a_3.
  5. Compute 1+2+3+4+5.
  1. 14.
  2. 48.
  3. a_5 = 2(5)+1 = 11.
  4. a_2=5, a_3=9.
  5. 15.

Subtopic guided practice and checkpoints

Sequence questions are about position, pattern, and rule choice. Always ask whether the pattern adds, multiplies, or depends on previous terms.

Lab 1: separate position from value

Guess first. In 4, 7, 10, 13, what is a_3?

Guided exercise.

The subscript is the position:

a_1 = 4
a_2 = 7
a_3 = 10
a_4 = 13

So a_3 = 10, not 3.

Checkpoint. For the sequence 9, 6, 3, 0, ..., identify a_1, a_2, and a_4.

Lab 2: use an explicit rule

Guess first. If a_n = 2n^2 - 1, is a_3 bigger than 10?

Guided exercise.

Substitute the position n = 3 into the rule:

a_3 = 2(3)^2 - 1
    = 18 - 1
    = 17

So yes, a_3 is bigger than 10.

Checkpoint. Compute a_5 for a_n = 3n - 4. Show the substitution step.

Lab 3: use a recursive rule

Guess first. If a_1 = 3 and a_{n+1} = 2a_n, is a_4 equal to 12 or 24?

Guided exercise.

A recursive rule must be applied one term at a time:

a_1 = 3
a_2 = 2a_1 = 6
a_3 = 2a_2 = 12
a_4 = 2a_3 = 24

Checkpoint. If a_1 = 5 and a_{n+1} = a_n - 2, find a_2, a_3, and a_4.

Lab 4: arithmetic, geometric, and sums

Guess first. Is 2, 6, 18, 54 arithmetic or geometric?

Guided exercise.

Arithmetic sequences add a constant difference. Geometric sequences multiply by a constant ratio. Here each term is multiplied by 3, so it is geometric.

For a finite sum, expand the notation before simplifying:

sum from i=1 to 4 of (2i) = 2 + 4 + 6 + 8 = 20

Checkpoint. Classify 10, 7, 4, 1 and 5, 15, 45, 135. Then compute sum from i=1 to 5 of i.

  1. For 9, 6, 3, 0, ..., a_1 = 9, a_2 = 6, and a_4 = 0.
  2. a_5 = 3(5) - 4 = 11.
  3. a_2 = 3, a_3 = 1, and a_4 = -1.
  4. 10, 7, 4, 1 is arithmetic with common difference -3. 5, 15, 45, 135 is geometric with common ratio 3. Also, 1 + 2 + 3 + 4 + 5 = 15.

Sequences checkpoint

Using this lesson with edumath and SymPy

Generate terms with Python.

terms = [3*n + 2 for n in range(1, 6)]
terms
[5, 8, 11, 14, 17]

Use edumath to create practice.

from edumath.discrete_math import sequence_next_term_exercise

exercise = sequence_next_term_exercise(seed=4)
exercise.prompt, exercise.expected
('Find the next term: 2, 6, 10, 14, ...', 18)

Use SymPy for finite sums.

import sympy as sp

i = sp.Symbol("i", integer=True, positive=True)
sp.summation(i, (i, 1, 5))

\(\displaystyle 15\)