Download Finding missing terms

Chapter 9: Sequences and Series
Generic symbol for a
TERM in a sequence
an
a1 = 1st term
a30 = 30th term
an = nth term
Example 1:
Find the first five terms of the given sequence:
Example 2:
Find the next 3 terms of the given sequence
an  5n  3
6, 1, -4, -9, …
9.2 Arithmetic Sequences
Example:
An arithmetic sequence is a sequence where the difference between
consecutive terms is constant. The difference between the
consecutive terms is called the common difference. (d)
4, 7, 10, 13,…
a1 = _____
d = _____
Example 1:
Is the sequence an arithmetic sequence?
If so, state the common difference.
A) 1, 4, 7, 10, …
B) 2, 4, 8, 16, …
C) 1, -5, -11, -17, …
1
Example 2:
Find the next 3 terms of the arithmetic sequence.
A) 12, 17, 22, …
B) -5, -1, 3, …
C) 12, -3, -18, …
WHAT IF YOU WERE ASKED:
What is the 11th term of the following arithmetic sequence?
6, 13, 20, …
TWO APPROACHES:
1. RECURSIVE: Continue the pattern until you reach the 11th number.
Arithmetic Sequence Formula
an = a1 + (n –1)d
2. EXPLICIT: Use a formula that represents the pattern
Example 3: Find a Specific Term
A) What is the 11th term of the arithmetic sequence 6, 13, 20, … ?
B) What is the 46th term of the arithmetic sequence
3, 5, 7, … ?
a1 = 1st term
an = nth term
n = subscript of an
(the counting number of the term)
d = the common difference
C) What is the 110th term of the arithmetic sequence
-5, -9, -13, …. ?
2
Example 4: Find a specific formula (pattern)
A) 2, 5, 8, 11, ….
Write a formula to represent the given sequence.
B) 10, 6, 2, -2, …
Example 5: Finding missing terms
HINT: Which variable do you need to know?
Find the missing numbers in each arithmetic sequence.
A) 80, ____, ____, 125, ….
an = a1 + (n –1)d
B) 146, ____, _____, ____, 78, ….
C) 35, _____, 53 , …
Arithmetic Mean:
In an arithmetic sequence, the middle term of any three
consecutive terms is the arithmetic mean (the average) of the
other two terms.
The number between x and y is
x y
2
Careful: This doesn’t work if you have more than 1 missing term! See example 5A and 5B!
Example 6:
A) Find the missing number 35, _____, 53 , …
B) Given that a5 = 15 and a7 = 59, find a6.
3
(Example 6 Continued)
C) The 9th and 11th terms of an arithmetic sequence
are 132 and 98, respectively. What is the 10th term?
D) Find the missing number 15, _____, 27
9.3 Geometric Sequences
Example:
An geometric sequence is a sequence where the ratio between
consecutive terms is constant. The ratio between the consecutive
terms is called the common ratio. (r)
4, 12, 36, 108…
a1 = _____
r = _____
Example 1:
Is the sequence a geometric sequence?
If so, state the common ratio, r.
A)
3, 12, 48, …
B) 16, 24, 36, …
C) 3, 6, 9, …
D)
5, 10, 50, ….
E) -8, 4, -2, 1, ….
F) 81 , 27 , 9 ,...
1
1
1
Example 2:
Find the next 3 terms of the geometric sequence.
A) 15, 30, 60, …
B) -120, 30, -7.5, …
C) 12, 18, 27, …
4
WHAT IF YOU WERE ASKED:
What is the 8th term of the following geometric sequence?
6, 12, 24, …
TWO APPROACHES:
1. RECURSIVE: Continue the pattern until you reach the 8th number.
Arithmetic Sequence Formula
an  a1  r n 1
2. EXPLICIT: Use a formula that represents the pattern
Example 3: Find a Specific Term
A) What is the 8th term of the arithmetic sequence 6, 12, 24, … ?
B) What is the 10th term of the geometric sequence
4, 12, 36, … ?
a1 = 1st term
an = nth term
n = subscript of an
(the counting number of the term)
r = the common ratio
C) What is the 7th term of the geometric sequence
-36, 18, -9, …. ?
D) What is the 8th term of a geometric sequence for which a1  3 and r  2 ?
5
an  a1  r n 1
Example 4: Find a specific formula (pattern)
Write a formula to represent the given sequence.
A) 2, 6, 18, ….
B) 10, 2, 0.4, …
Example 5: Finding missing terms
HINT: Which variable do you need to know?
an  a1  r n 1
Find the missing numbers in each geometric sequence.
A) 2, ____, ____, -54, ….
B) 9, ____, _____, ____, 144, ….
Geometric Mean:
In a geometric sequence, the middle term of any three
consecutive terms is the geometric mean of the other two terms.
C) 28, _____, 7 , …
The number between x and y is  xy
Careful: This doesn’t work if you have more than 1 missing term! See example 5A and 5B!
Example 6:
A) Find the missing number 28, _____, 7 , …
B) Given that a5 = 5 and a7 = 2.8125, find a6.
C) The 9th and 11th terms of a geometric sequence
are -8 and -2, respectively. What is the 10th term?
D) Find the missing number 16, _____, 9
6
9.4 Arithmetic Series
An arithmetic series is a sum of the terms in an arithmetic sequence (see lesson 9.2)
Example 1:
Find the sum of the first 100 positive integers.
Example 2: Finding Finite Sums
FINITE Arithmetic Series Formula
A) What is the sum of the arithmetic series where
a1 = 7 , an = 79, and n = 8 ?
Sn 
n
 a1  an 
2
a1 = 1st term
an = nth term (the last term in the series)
n = subscript of an
(the counting number of the term)
Sn = the Sum of the n terms in the series
B) What is the sum of the arithmetic series where
an = 80, n=11, and d = 7 ?
9.2 Arithmetic Sequence: an = a1 + (n –1)d
7
C) What is the sum of the arithmetic series where
14 + 17 + 20 + … + 116 ?
D) What is the sum of the arithmetic series where
20 + 18 + 16 + … + -24 ?
Summation Notation
You can use the Greek capital letter sigma Σ to indicate a sum. With it, you use limits to indicate how many terms
you are adding. Limits are the least and greatest values of n in the series. You write the limits below and above
the Σ to indicate the first and last terms of the series.
10
 3n
last value of n
formula for the terms in the series
n 1
first value of n
Write out this arithmetic series and find the sum.
Example 3: Finding Sum from Summation Notation
What is the sum of the given series?
A)
40
Strategy:
n 1

Find a1 (the 1st term)
Plug the lower limit into the formula for
the nth term.

Find an (the last term)
Plug the upper limit into the formula for
the nth term.

Find n (the number of terms)
n = upper limit – lower limit + 1

Find the SUM
 (3n  8)
Use Sn 
n
 a1  an 
2
8
B)
50
 (4n  7)
n 1
C)
12
 (2n)
n4
Example 4: Vocabulary Review
Draw a line to match the
word/phrase in column A
with the correct definition
in column B.
9.5 Geometric Series
A geometric series is a sum of the terms in a geometric sequence (see lesson 9.3)
Example 1:
(from your textbook pg 596)
According to the story, what would the first 5 terms of this series be?
Is this an arithmetic series or a geometric series? How can you tell?
9
Use the Geometric Series formula at the right.
How many kernels of wheat did the soldier
request?
FINITE Geometric Series Formula
n
1
n
S 
a (1  r )
1 r
or
Sn 
a 1 an r
1 r
a1 = 1st term
an = nth term (the last term in the series)
n = subscript of an
(the counting number of the term)
r = the common ratio
Sn = the Sum of the n terms in the series
Example 2: Sums of Finite Geometric Series
Find the sum of the finite geometric series with the following information
A) a1 = -15, r = -2, and n = 6
B)
a1=81, r = 1 , n = 5
3
C) 4 + 12 + 36 +…+ 2916
D) –6 + 18 –54 + …+ 13122
10
Infinite Geometric Series
Think about the following infinite geometric series:
4 1
1 1 1
1
  
 ...
4 16 64 256
What are the following “Partial Sums”?
1 1
1
1
 

4 16 64 256
S1 = 4
S6 =
S2 = 4  1
S7 = 4  1  1  1  1  1  1
S3 =
4 1
4 1
4 16
1
4
S8 =
1 1
4 16
S4 = 4  1  
1 1
1

4 16 64
S5 = 4  1  
4 1
64
256 1024
1 1 1
1
1
1
  


4 16 64 256 1024 4096
1 1
1
1
1
1
1





4 16 64 256 1024 4096 16384
S9 = 4  1  
Sn 
What appears to be happening?
In an infinite geometric series where r  1 , one of two things can happen:
Case 1:
r  1 or r  1
Case 2:
1  r  1
Terms GROW rapidly.
Sum grows RAPIDLY.
S approaches  or  .
“DIVERGES”
Terms DIMINISH rapidly.
Sum starts growing by negligible amounts.
S approaches a finite sum. (an actual number!)
“CONVERGES”
11
Example 3: Sums of Infinite Geometric Series
INFINITE Geometric Series Formula
Does the infinite geometric series converge or diverge?
If it converges, find the sum.
S
A) 18  9  4.5  ...
a1 = 1st term
r = the common ratio
S = the Sum of the series
B) 18  6  2  ....
D)
81  27  9  ...
C) 1 
E)
a1
1 r
if
1  r  1
5 25

 ....
4 16
0.5  0.52  0.5408  ...
Example 4: Repeating Decimals  Fractions
Think about
0.52
12