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Course: Algebra (all content) > Unit 4

Lesson 2: Constructing arithmetic sequences
Math
Algebra (all content)
Sequences
Constructing arithmetic sequences
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Explicit formulas for arithmetic sequences

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Learn how to find explicit formulas for arithmetic sequences. For example, find an explicit formula for 3, 5, 7,...
Before taking this lesson, make sure you are familiar with the basics of arithmetic sequence formulas.

How explicit formulas work

Here is an explicit formula of the sequence 3,5,7,…
a(n)=3+2(n−1)
In the formula, n is any term number and a(n) is the nth term.
This formula allows us to simply plug in the number of the term we are interested in, and we will get the value of that term.
In order to find the fifth term, for example, we need to plug n=5 into the explicit formula.
a(5)=3+2(5−1)=3+2⋅4=3+8=11
Cool! This is in fact the fifth term of 3,5,7,…

Check your understanding

1) Find b(10) in the sequence given by b(n)=−5+9(n−1).
b(10)=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Writing explicit formulas

Consider the arithmetic sequence 5,8,11,… The first term of the sequence is 5 and the common difference is 3.
We can get any term in the sequence by taking the first term 5 and adding the common difference 3 to it repeatedly. Check out, for example, the following calculations of the first few terms.
nCalculation for the nth term
15=5+0⋅3=5
25+3=5+1⋅3=8
35+3+3=5+2⋅3=11
45+3+3+3=5+3⋅3=14
55+3+3+3+3=5+4⋅3=17
The table shows that we can get the nth term (where n is any term number) by taking the first term 5 and adding the common difference 3 repeatedly for n−1 times. This can be written algebraically as 5+3(n−1).
In general, this is the standard explicit formula of an arithmetic sequence whose first term is A and common difference is B:
A+B(n−1)

Check your understanding

2) Write an explicit formula for the sequence 2,9,16,….
d(n)=

3) Write an explicit formula for the sequence 9,5,1,….
e(n)=

4) The explicit formula of a sequence is f(n)=−6+2(n−1).
What is the first term of the sequence?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
What is the common difference?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Equivalent explicit formulas

Explicit formulas can come in many forms.
For example, the following are all explicit formulas for the sequence 3,5,7,…
  • 3+2(n−1) (this is the standard formula)
  • 1+2n
  • 5+2(n−2)
The formulas may look different, but the important thing is that we can plug an n-value and get the correct nth term (try for yourselves that the other formulas are correct!).
Different explicit formulas that describe the same sequence are called equivalent formulas.

A common misconception

An arithmetic sequence may have different equivalent formulas, but it's important to remember that only the standard form gives us the first term and the common difference.
For example, the sequence 2,8,14,… has a first term of 2 and a common difference of 6.
The explicit formula 2+6(n−1) describes this sequence, but the explicit formula 2+6n describes a different sequence.
In order to bring the formula 2+6(n−1) to an equivalent formula of the form A+Bn, we can expand the parentheses and simplify:
=2+6(n−1)=2+6n−6=−4+6n
Some people might prefer the formula −4+6n over the equivalent formula 2+6(n−1), because it's shorter. The nice thing about the longer formula is that it gives us the first term.

Check your understanding

5) Find all correct explicit formulas of the sequence 12,7,2,…
Choose all answers that apply: وہ سب سلیکٹ کریں جو مناسب ہے

Challenge problems

6*) Find the 124th term of the arithmetic sequence 199,196,193,…
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

7*) The first term of an arithmetic sequence is 5 and the tenth term is 59.
What is the common difference?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

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