Download Sequences and Series!!!

Sequences
and Series!!!
Finding the Degree
of a Sequence
Begin by finding the difference between
adjacent numbers
2, 5, 8, 11, 14
^
3
^
3
^
3
^
3
3, 10, 29, 66, 127, 218
^
7
^ ^
19 37
^
61
^
91
If the number you add each time is the same, you
are done. YAY!!!
If the number you add each time is different,
repeat the process- find the difference between
the differences. Continue until they are the same.
3, 10, 29, 66, 127, 218
^
7
^ ^
^
^
19 37 61
91
^
^ ^
^
12 18 24
30
^
^
^
6
6
6
If you complete this step….
• One time, the sequence is linear.
• Two times, the sequence is quadratic.
• Three times, the sequence is cubic.
• Four times, the sequences is quartic.
Finding the nth term of a
linear sequence
1. Find the difference between the numbers.
2. Find out what you would have to add or subtract to
your difference, in order to get your start value.
3. Write your formula:
(difference)*n + (the number you add/subtract)
Example
6, 8, 10, 12, 14
Difference: 2
Add: 4
nth term: 2n + 4
Example
4, 1, -2, -5, -8
Difference: -3
Add: 7
nth term: -3n + 7
Part 2
Finding the nth term of a
quadratic sequence
Note that ANY 2nd degree polynomial can be written in the Quad. Equation,
y = ax2 + bx + c
ANY quadratic will have this form, though sometimes b or c is 0.
examples: 5x2 – 2x + 3, x2 – 4x,
-3x2 + 7
What ARE a, b, & c in each of these quadratic expressions?
input n 1
output f(n) 6
2 3 4 5
19 42 75 118
Our problems start with a chart showing the sequence f(n) for each value of n.
and we need to find the nth term
So, let’s change the quadratic equation to match our problem:
y = ax2 + bx + c
changes to
f(n) = an2 + bn + c
or an2 + bn + c = f(n)
an2
+ bn + c = f(n)
input n 1
output f(n) 6
STEP
A) Write 3 equations by replacing n in the
above equation with 1, then 2, then 3.
Set each equation equal the f(n) from
the chart.
B) Simplify each equation
2 3 4 5
19 42 75 118
Example .
#1
a(12) + b(1) + c = 6
#2
a(22) + b(2) + c = 19
#3
a(32) + b(3) + c = 42
#1
a + b + c = 6
#2
4a + 2b + c = 19
#3
9a + 3b + c = 42
#1
a + b + c = 6
#2
4a + 2b + c = 19
#3
9a + 3b + c = 42
input n 1
output f(n) 6
Example:
STEP
C) Subtract equation #3 – #2 
and
subtract equation #2 – #1,

D) Then subtract those two answers
from each other.
2 3 4 5
19 42 75 118
9a + 3b + c = 42
– (4a + 2b + c = 19)
5a + b
= 23
4a + 2b + c = 19
– ( a + b + c = 6)
3a + b = 13
5a + b = 23
– (3a + b = 13)
2a = 10
STEP
Find a
input n 1
output f(n) 6
2 3 4 5
19 42 75 118
example:
2a = 10  a = 5
E) Using the equation 3a + b = 13
substitute 5 in for a
F) Now sub a & b into equation #1
and find c
G) Write your final equation using a, b, and c into
the quadratic equation. (Check your answer.)
3(5) + b = 13
b = -2
#1  a + b + c = 6
5 + -2 + c = 6
c=3
a = 5, b = -2, c = 3
so the nth term is:
5n2 – 2n + 3
1
2
3
Practice:
4
5 6
7
8, 15, 24, 35, 48, 63, 80
Put in the n’s to go with this sequence
(click for answer) 
nth term for
the quadratic:
n2 + 4n + 3
Example:
-8, 11, 42, 85, 140, 207, 286
(click for answer) 
nth term for
the quadratic:
6n2 + n – 15
END QUADRATICS
Finding the nth term of a
cubic (or quartic)
Use more equations to find
a b c & d!!! 
Arithmetic vs. Geometric Sequences
Arithmetic is just another name for a linear sequence.
Geometric is where the ratio (what you multiply)
between consecutive terms is constant. The number
you multiply by is called the common ratio.
Finding the nth term of a
geometric sequence
Use this formula!
(𝑛−1)
𝑡1 ∗ 𝑟
• 𝑡1 = first term
• r = common ratio
Series vs. Sequences
A sequence is just a list of numbers that follow a
pattern.
A series is like a sequence, but the numbers are added
together.