F-IF.3
Vocabulary
Sequence: a set of numbers in a specific order
Terms: the numbers in the sequence
Arithmetic sequence: is a numerical pattern that increases or decreases at a constant rate or value
called the common difference
Common difference: difference between the terms in a sequence
Geometric Sequences: numerical pattern that increase or decreases by a common ratio
Arithmetic Formulas
Recursive Formulas (Used to find the next term in a sequence)
The recursive formula for an arithmetic sequence is written in the form:
where 𝑎𝑛 is the next number you are trying to find in the sequence, 𝑎𝑛−1 is the last number you have
in the sequence, and d is the common difference (how much the numbers in the sequence is going
up or down by)
This equation is better understood by the students as: NEXT = NOW +/- C,
where NEXT is the next number in the sequence you are trying to find, NOW is the last number you
have in the sequence, and C is the common difference. This is how common core wants you to
teach it as.
Example:
1. What is the recursive formula for the sequence: 15, 12, 9, 6…
NEXT = NOW +/- C
NEXT = NOW – 3
(because the numbers are decreasing by 3 each time)
(This formula written in the
form would be 𝑎𝑛 = 𝑎𝑛−1 − 3)
2. Use the recursive formula to find the next term in the sequence.
NEXT = NOW – 3
NEXT = 6 – 3
NEXT = 3
Explicit Formulas(Used to find the 𝑛𝑡ℎ term in a sequence)
The recursive formula is only valauble if you are trying to find the very next term in a sequence. It
is not beneficial it you want to find, say the 50th term. For this you need to find the explicit form.
The explicit form is written in the form:
, where 𝑎𝑛 is the term you are trying to
find, 𝑎1 is the first term in the sequence, n is the numbered term you are trying to find, and d is the
common difference. This formula will look like y = mx + b once all numbers are put in and
simplified. All student have to do is substitute the numbers into the correct place in the formula and
simplify.
Example:
1. Find the explicit formula for the sequence: 15, 12, 9, 6…
𝑎1 = 15
d = -3
𝑎𝑛 = 15 + (𝑛 − 1)(−3)
𝑎𝑛 = 15 - 3n + 3
𝑎𝑛 = -3n + 18
2. Find the explicit formula to find the 50th term in the sequence above.
n = 50
𝑎𝑛 = -3n + 18
𝑎𝑛 = -3(50) + 18
𝑎𝑛 = -150 + 18
𝑎𝑛 = -132
Geometric Sequences
Recursive Sequences
The formula for the recursive geometric formula is:
where𝑎𝑛 is the next number you are trying to find, 𝑎𝑛−1 is the last number you have in the
sequence, and r is the common ratio. Common core uses the formula: NEXT = B ∙ NOW where B is
the base in an exponential formula.
Examples:
1. Find the recursive formula for 0.4, 0.04, 0.004, 0.0004, . . .
NEXT = B ∙ NOW
NEXT = 0.1 ∙NOW
Each number in the sequence is being multiplied by 0.1
2. Use the recursive formula to find the next term in the sequence 0.4, 0.04, 0.004, 0.0004, . . .
NEXT = 0.1 ∙NOW
NEXT = 0.1∙ .0004
NEXT = .00004
Explicit Formulas
Once again, explicit formulas need to be formulated in order to find any term in the sequence not
just the next one. The form of this equation is
, where 𝑎𝑛 is the term you are trying to
find, 𝑎1 is the first term in the sequence, r is the common ratio, and n is the numbered term you are
trying to find. The common core unpacking documents do not have their own formula for this
equation.
Examples:
1. Find the explicit formula for the sequence: 0.4, 0.04, 0.004, 0.0004, . . .
𝑎1 = 0.4
d = 0.1
𝑎𝑛 = 0.4 ∙ 0.1𝑛−1
2. Find the explicit formula to find the 50th term in the sequence above.
n=7
𝑎𝑛 = 0.4 ∙ 0.1𝑛−1
𝑎𝑛 = 0.4 ∙ 0.17−1
𝑎𝑛 = 0.4 ∙ 0.16
𝑎𝑛 = 0.4 ∙ 0.000001
𝑎𝑛 = 0.0000004
Activity:
What Comes Next? And Next?And Next?
1. 3, 6, 9, 12, …
2. 45, 39, 33, 27, …
3. 4, 5, 8, 13, 20, … 4. -12, -7, -2, 3, …
5. 1, 2, 4, 7, …
6. -2, -1.75, -1.5, -1.25, …
Writing an Arithmetic Sequence
(This worksheet can be used to teach how to form the explicit equations)
Each term in an arithmetic sequence can be expressed in terms of the first term (a1)
and the common difference (d).
Number Example: 8, 11, 14, 17…
Term
Symbol
In terms of a1 and d
First term
𝑎1
𝑎1
Second term
𝑎2
𝑎1 + 𝑑
Third term
𝑎3
𝑎1 + 2𝑑
Fourth term
𝑎4
𝑎1 + 3𝑑
nth term
𝑎𝑛
𝑎1 + (𝑛 − 1)𝑑
Numbers
8
The equation for finding the nth term of an arithmetic sequence is: _____________
Where
𝑎𝑛 :
𝑎1 :
n:
d:
Practice:
Find the nth term…
1. Find the 14th term in the arithmetic sequence 9, 17, 25, 33…
2. Find the 10th term of the sequence 7, 14, 21, 28, …
Write the equation for the sequence…
3. Write the equation for the nth term in the sequence 12, 23, 34, 45, ….
4. Write the equation for the nth term in the sequence -8, 1, 10, 19, …
(The following worksheets can be used as group/partner assignments or even
stations)
Find an Equation…
Jacob is training for a marathon. The table below shows the total number of miles he
has run each week.
Week
1
2
3
4
5
Number of Miles
9
15
21
27
33
1. Do the number of miles ran form an arithmetic sequence? Justify your answer.
2. Write an equation for the sequence.
3. How many miles will Jacob run during his 8th week of training?
4. Graph the sequence.
Find an Equation…
The table below shows the total number of rotations a Ferris wheel turns during a
passenger’s ride at a carnival.
Number of Minutes 1
2
3
4
5
Number of Rotations 2
9
16
23
30
1. Do the number of rotations form an arithmetic sequence? Justify your answer.
2. Write an equation for the sequence.
3. How many times will the Ferris wheel turn in 8 minutes?
4. Graph the sequence.
Find an Equation…
The table below shows the number of calories a person consumes each day at the
beginning of a new diet.
Number of Days
1
2
3
4
5
Number of Calories 2500 2375 2250 2125 2000
1. Do the number of calories form an arithmetic sequence? Justify your answer.
2. Write an equation for the sequence.
3. How many calories will the dieter consumer on the 8th day of their diet?
4. Graph the sequence.
Find an Equation…
The table below shows the number of miles Natasha rode her bike each week while
training for a marathon.
Number of Weeks
1
2
3
4
5
Number of Miles
5
11
17
23
29
1.Do the number miles form an arithmetic sequence? Justify your answer.
2. Write an equation for the sequence.
3. How many miles will she have rode on the 8th day?
4. Graph the sequence.
Name ________________
Arithmetic Sequences
Exit Ticket
Find the nth term in the sequence. Use the equation, 𝑎1 + (𝑛 − 1)𝑑.
1) a1 =8, d = 3, n = 16
2) a1 =5, d = 5, n = 25
Find the equation for the sequence:
3) 9, 13, 17, 21…