Sequence Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Sequence.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An ordered list of numbers generated by a rule, where each number has a specific position (first, second, third,...).

A pattern of numbers: first term, second term, third term, and so on.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A sequence assigns a term to each position $1,2,3,\ldots$ by a generating rule; terms are listed, not added.

Common stuck point: The procedure for sequence is the easy part; the trap is treating a sequence as a sum. Asking "Am I listing terms by position ( $a_n$ ) rather than adding them up?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I listing terms by position ( $a_n$ ) rather than adding them up?

Worked Examples

Example 1

easy

Write the first five terms of the sequence

a_n = \frac{n}{n+1}

for

n = 1, 2, 3, \ldots

Answer

\frac{1}{2},\; \frac{2}{3},\; \frac{3}{4},\; \frac{4}{5},\; \frac{5}{6}

First step

a_1 = \frac{1}{2}

Full solution

2
$a_2 = \frac{2}{3}$ .
3
$a_3 = \frac{3}{4}$ .
4
$a_4 = \frac{4}{5}$ .
5
$a_5 = \frac{5}{6}$ .

Each term is found by substituting

n

into the formula. The terms increase and approach 1, illustrating that this sequence converges to 1 as

n \to \infty

Example 2

medium

Determine whether the sequence

a_n = \frac{3n^2 + 1}{n^2 + 2}

converges or diverges. If it converges, find the limit.

Example 3

easy

Show step-by-step output for:

n=2

;

n=n*3

;

n=n+1

; OUTPUT

n

Example 4

medium

A recipe lists: (1) bake 30 min, (2) preheat oven, (3) mix batter. Why is order (2),(3),(1) correct rather than (1),(2),(3)?

Example 5

medium

A program reads two numbers, prints their sum, then their product. Write the sequence of 4 steps in plain pseudocode order.

Example 6

hard

A deploy script runs: build, run tests, push to prod. A junior reorders to: push, build, test. List 2 concrete failure modes.

Example 7

challenge

Given 4 statements

S_1..S_4

where

S_3

depends on

S_1

and

S_4

depends on

S_2

, how many valid total orderings exist?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Write the first four terms of

b_n = (-1)^n \cdot \frac{1}{n}

Example 2

medium

Determine whether

a_n = \frac{2^n}{n!}

converges or diverges.

Example 3

easy

3, 7, 11, 15, \ldots

a sequence or a series?

Example 4

easy

Find the 4th term of the sequence

a_n=2n+1

Example 5

easy

List the first three terms of

a_n=n^2

Example 6

easy

What is the next term in

5, 10, 20, 40, \ldots

Example 7

easy

Does the sequence

a_n=1-\frac{1}{n}

increase or decrease as

n

grows?

Example 8

easy

Write the general term for

3, 5, 7, 9, \ldots

starting at

n=1

Example 9

easy

In the sequence

a_n=\frac{n}{n+1}

, which term equals

\frac{4}{5}

Example 10

easy

2, 2, 2, 2, \ldots

a valid sequence? What is its limit?

Example 11

medium

A sequence is defined recursively by

a_1=3

a_{n+1}=a_n+4

. Find

a_5

Example 12

medium

The general term is

a_n=\frac{2n-1}{3n+2}

. What value do the terms approach as

n\to\infty

Example 13

medium

Find a formula for the

n

th term of

1, 4, 9, 16, 25, \ldots

and use it to find the 8th term.

Example 14

medium

The sequence

a_n=(-1)^n

gives

-1,1,-1,1,\ldots

. Does it converge?

Example 15

medium

Given

a_n=a_{n-1}\cdot\frac{1}{2}

with

a_1=16

, find

a_4

and describe long-term behavior.

Example 16

medium

For the sequence

a_n=2n+1

, what is the difference

a_{n+1}-a_n

Example 17

challenge

A sequence satisfies

a_1=1

and

a_{n+1}=\frac{a_n+2}{a_n+1}

. Show its terms approach

\sqrt{2}

Example 18

challenge

The Fibonacci sequence is

1,1,2,3,5,8,\ldots

with

F_{n+1}=F_n+F_{n-1}

. Show

\frac{F_{n+1}}{F_n}

approaches

\frac{1+\sqrt5}{2}

Example 19

challenge

Prove the sequence

a_n=\frac{3n^2+1}{n^2+2}

is bounded above by 3 and find its limit.

Example 20

medium

Find the 10th term of the sequence defined by

a_n=3\cdot2^{n-1}

Example 21

medium

A sequence has

a_n=\frac{n^2}{n^2+1}

. Is it increasing, and what is its limit?

Example 22

medium

Find the general term of

2, 6, 12, 20, 30, \ldots

(hint: products of consecutive integers).

Example 23

easy

Trace:

x=7

;

y=x-2

;

z=y+1

. What is

z

Example 24

easy

Trace:

a=4

;

b=a+a

;

c=b+a

. What is

c

Example 25

easy

Order the steps to send an email: (1) Click Send, (2) Type message, (3) Open email app, (4) Enter recipient. Give the correct order.

Example 26

easy

Trace:

p=3

;

q=p

;

p=p+10

. What is

q

Example 27

easy

Trace:

s

= '';

s = s + 'g'

;

s = s + 'o'

. What is

s

Example 28

medium

Trace:

x=10

;

y=x/2

;

x=x-y

; OUTPUT

x

Example 29

medium

Order so that account balance prints correctly: (1) print balance, (2) deposit = 50, (3) balance = balance + deposit, (4) balance = 100. Give the order.

Example 30

medium

Trace:

a=6

;

b=4

;

a=a+b

;

b=a-b

; OUTPUT

a

b

Example 31

medium

Trace:

m=12

;

m=m\%5

;

m=m*4

; OUTPUT

m

Example 32

medium

Trace:

L=[1,2]

;

L

.append(3);

L

.append(

L

[0]); OUTPUT

L

Example 33

medium

Trace:

a=2

;

b=a*a

;

c=b*a

;

d=c+b+a

; OUTPUT

d

Example 34

medium

Trace:

x=5

;

y=10

; swap (no temp):

x=x+y

;

y=x-y

;

x=x-y

. Final

x,y

Example 35

medium

Trace:

count=0

;

count=count+1

;

count=count*2

;

count=count+1

; OUTPUT

count

Example 36

medium

If you reorder `open file; read file; close file` to `read; open; close`, what happens at runtime?

Example 37

hard

Trace:

a=1

;

b=2

;

c=3

;

a=b

;

b=c

;

c=a

. Final

a,b,c

Example 38

hard

A statement using

z

appears at line 3;

z

is assigned at line 7. Will the program work as written? Why or why not?

Example 39

hard

Trace:

x=2

;

y=x

;

x=x+1

;

z=x+y

;

y=z-x

; OUTPUT

x,y,z

Example 40

challenge

Trace:

a=1

;

b=a

;

a=a+b

;

b=a+b

;

a=a+b

; OUTPUT

a

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Related Concepts

Arithmetic Sequence Geometric Sequence Series