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Sequences and Series
Arithmetic Sequences
A sequence is an ordered list of numbers.
An arithmetic sequence is an ordered list that results from
adding the same number each time.
Examples:
3, 7, 11, 15, … (plus 4 each time)
19, 26, 33, 40, … (plus 7 each time)
5, 2, -1, -4, … (plus -3 each time)
The Common Difference
For any arithmetic sequence, the common difference is
the fixed amount that is added to each term in the
sequence.
Example: 1, 3, 5, 7, …
the common difference, d, is +2. So we write d=2.
Find the next three terms in the
sequence and identify the common
difference…
1. 1, 6, 11, __, __, __
2. 7, 19, 31, __, __, __
d= __
d= __
3. 148, 152, 156, __, __, __ d= __
4. 88, 81, 74, __, __, __
d= __
Check your answers…
1. 1, 6, 11, _16_, _21_, _26_
d= _5_
2. 7, 19, 31, _43_, _55_, _67_
d= _12_
3. 148, 152, 156, _160_, _164_, _168_ d= _4_
4. 88, 81, 74, _67_, _60_, _53_
d= _-7_
Connection from Math 1
In Math 1, we talked about NOW-NEXT
rules.
For the pattern…. 5, 8, 11, 14… we would
write NEXT=NOW + 3, starting at 5.
This is the INFORMAL way to describe
sequences.
A little more challenging…
Find the 10th term in the sequence:
-7, 2, 11, …
Did you get 74?
You can continue the pattern until you
get to the 10th number, and that is
pretty easy.
Find the 78th term in the
sequence:
5, 11, 17, ….
Um…yikes! This would take
a long time.
Hopefully there is a shortcut!
Explicit Rules & Recursive
Rules
Recursive Rules
A NOW-NEXT rule is an informal recursive rule. The
prefix “re” means again and is used to indicate
repetition.
REcursive…therefore needs to talk about repeating from
one to another.
Arithmetic
Recursive Rule:
an is the nth term.
an-1 is the previous term.
d is the common difference.
Recursive Rule Examples
3, 7, 11, 15, 19, 23, ….
!
The first term is 3 and we can write a1=3
The second term is 7 and we write a2=7
The third term is 11 and we write a3=11
The fourth term is 15 and we write a4=15
The 5th term is 19 and we write a5=19
The 6th term is 23 and we write a6=23
!
If we keep going infinitely many times… we can let n
represent the number term we want. Thus we get an
Writing a recursive
rule:
Look for the pattern, just fill in d.
!
Ex: 18, 20, 22, 24, …
!
The pattern is +2 each time so d=2.
We fill in our rule for d and we get:
!
an=an-1 + 2
Practice w/ Recursive
Rules:
1. 9, 17, 25, 33, …
!
2. -4, -7, -10, -13, …
!
Write a recursive rule for both.
Answers:
1. 9, 17, 25, 33, …
!
!
2. -4, -7, -10, -13, …
!
Explicit Rule!!!
This is the most important rule (and another formula to
memorize).
!
An explicit rule is one that will work to calculate ANY term
in the sequence.
!
These are similar to creating a y= rule for a table to find y
for any given value of x.
Arithmetic Explicit Rule
The rule lets you figure out what the 78th term
in the sequence is as long as you know the
first term (a1) and the common difference (d).
Example:
5, 8, 11, 14, …
(we know it is +3 each time, so d=3)
(we know the first term is 5, so a1=5
So when I plug things into the rule
Explicit:
Simplify…
Practice writing an Explicit
Rule:
-2, -5, -8, …
Find the common difference, d.
Plug in a1 and d into the explicit rule formula
Simplify:
DO THIS in your notes now.
Answer:
Finding the nth term:
Write an equation for the nth term of the sequence
-2, 5, 12, 19, 26
Now, Find the 25th term (n=25)
ANSWERS:
Rule:
25th term:
Finding missing terms
(arithmetic means) in sequence…
Consider: -7, ____, _____, _____, _____, _____, 11
Need to find missing terms but look at what we know:
a1 = -7 (first term) and a7=11 (seventh term)
What we don’t know is the common difference.
So we plug in a1 = -7, a7=11, and n=7 to find d.
Continued:
So we found the common difference to be 3 so then
we can use this to fill in the blanks. This means that
the sequence goes up 3 each time. So starting at -7,
up 3 will be -4. Fill in the rest of the blanks.
-7, _-4_, ___, ___, ___, ___, 11
Answer:
-7, -4, -1, 2, 5, 8, 11
Finding out which term is a certain value:
Example: Which term is 763 in the arithmetic
sequence -7, 4, 15, 26, …
•
•
•
•
Think about what we do know…
763 is some unknown term (an)
The first term is -7 so a1=-7
The common difference is 11, so d=11
PLUG the info we know into the formula and solve
for what we don’t know (n).
Finding the first term (if unknown)
a101=100, d=7
This means that an=100 and n=101 and d=7. The only
thing we are missing is a1
Plug these in and solve for a1
Try the following on your own:
1.
6, 12, 18, … Find the 42nd term.
!
!
2.
5, 11, 17, … Find the 78th term.
!
!
3.
10, __, __, __, __, __, __, __, __, -62
!
!
4. Which term is -23 in the sequence: 25, 21, 17, 13, …