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Wize High School Grade 11 Math Textbook > Sequences & Series

Arithmetic Sequences

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What is a Sequence?

A sequence is a list of numbers that follow some kind of order/ rule/ pattern.

Examples
  • S1={1,2,3,4,5}S_1=\{1,2,3,4,5\} is a sequence
  • S2={−2,1,4,7,10,13,...}S_2=\{-2,1,4,7,10,13,...\} is a sequence
  • S3={2,4,8,16,32,...}S_3=\{2, 4, 8, 16, 32, ...\} is a sequence

The length of a sequence

The sequence can go on forever (infinite sequence) or it can end at some point (finite sequence)


Terms in a sequence

Each number inside a sequence is usually called a term.
  • The first number in a sequence is called the first term
  • The second number in a sequence is called the second term
  • ...
  • If the sequence is finite, then the last number in a sequence is called the last term
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Arithmetic Sequence


An arithmetic sequence is a special type of sequence where the numbers are either getting smaller or larger by the same amount.
  • If you take any number in the sequence and subtract the previous number, you'll get the same constant result
  • We call this constant result the common difference dd

Examples
  • Is S1={1,2,3,4,5}S_1=\{1,2,3,4,5\} an arithmetic sequence?
    yes
  • What is the common difference?
    1
  • Is S2={−2,1,4,7,10,13,...}S_2=\{-2,1,4,7,10,13,...\} an arithmetic sequence?
    yes
  • What is the common difference?
    3
  • Is S3={1,−1,−3,−5,−7,...}S_3=\{1,-1,-3,-5,-7,...\} an arithmetic sequence?
    yes
  • What is the common difference?
    -2
  • Is S4={2,4,8,16,32,...}S_4=\{2,4,8,16,32,...\} an arithmetic sequence?
    no
  • What is the common difference?
    the terms are not changing by a constant amount -- no common difference
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General Term

If the first term is a1a_1 and the common difference is dd, then the arithmetic sequence looks like this:
a1=a1a2=a1+da3=a1+2da4=a1+3d⋮\large{\begin{array}{rcl} a_1&=&a_1\\ a_2&=&a_1+d\\ a_3&=&a_1+2d\\ a_4&=&a_1+3d\\ &\vdots\\ \end{array}}
The general term is a formula for any particular term within the arithmetic sequence:
an=a1+(n−1)d\Large\boxed{a_n=a_1+(n-1)d}

Practice: Arithmetic Sequence

Given the sequence 4, −1, −6, −11, ...4,~-1,~-6,~-11,~...

a) determine whether it is an arithmetic sequence.
b) determine the next term in the sequence.
c) determine the formula for the general term for this sequence.

Practice: Arithmetic Sequence

Given the arithmetic sequence 12, 56, 76, 32, ...\dfrac{1}{2},~\dfrac{5}{6},~\dfrac{7}{6},~\dfrac{3}{2},~...

a) find the common difference.
b) determine the next term in the sequence.
c) determine the formula for the general term for this sequence.

Practice: Arithmetic Sequence as a Relation

What type of relation does an arithmetic sequence exhibit?

Practice: Arithmetic Sequence

On June 1st, 2000, Anthony deposited a certain amount into his savings account. Then he has been depositing $500 into his savings account on the first day of every month thereafter. If on Feb 5th, 2001, Anthony's saving account had $5,700, answer the following questions.

a) How much did Anthony first deposit on June 1st, 2000?

b) How much money will Anthony have in his savings account on Feb 10th, 2002?

c) Write the general term formula that represents the amount of money in Anthony's savings account nn months after June 1st, 2000.


Summary - Arithmetic Sequence

Consecutive terms in an arithmetic sequence differ by the same common difference (can be a positive or negative number). So, to get the next number in an arithmetic sequence, we add this common difference to the current number.

We can use this formula to find the nth term in an arithmetic sequence:

an=a1+(n−1)d\Large\boxed{a_n=a_1+(n-1)d}
where
  • ana_n is the nth term in the arithmetic sequence
  • a1a_1 is the first term in the arithmetic sequence
  • dd is the common difference between consecutive terms