Chapter 8 – Sequences, Series, and the
Binomial Theorem
Section 1
Section 2
Section 3
Section 4
Sequences and Series
Arithmetic Sequences and Partial Sums
Geometric Sequences and Series
The Binomial Theorem
Vocabulary
Infinite sequence
Finite sequence
Recursively
Factorial
Summation (Sigma) notation
Series
Arithmetic sequence
Finite arithmetic sequence
Geometric sequence
(Infinite) Geometric series
Binomial coefficients
The Binomial Theorem
Pascal’s Triangle
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Section 8.1 Sequences and Series
Objective:
In this lesson you learned how to use sequence, factorial, and summation notation to
write the terms and sums of sequences.
Important Vocabulary
Infinite Sequence
Finite Sequence
Summation (Sigma) Notation
Series
Recursively
Factorial
Arithmetic Sequence
Finite Arithmetic Sequence
I.
Sequences
An infinite sequence is:
What you should learn:
How to use sequence notation
to write the terms sequences
The function values 𝑎1 , 𝑎2 , 𝑎3 , 𝑎4 , … , 𝑎𝑛 , … are the ____________ of an infinite sequence.
A finite sequence is:
To find the first three terms of a sequence, given an expression for its 𝑛th term, you:
To define a sequence recursively, you need to be given
________________________________________________. All other terms of the sequence are then
defined using ________________________.
II.
Factorial Notation
If 𝑛 is a positive integer, 𝒏 factorial is defined by
______________________________________________.
What you should learn:
How to use factorial notation
Zero factorial is defined as _________.
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III.
Summation Notation
The sum of the first 𝑛 terms of a sequence is represented by
the summation or sigma notation,
𝑛
∑ 𝑎𝑖 =
What you should learn:
How to use summation
notation to write sums
𝑖=1
where 𝑖 is called the ______________________________, 𝑛 is the
______________________________, and 1 is the _____________________________________.
Properties of Sums:
1.
2.
3.
4.
Sums of Powers of Integers:
1. 1 + 2 + 3 + 4 + ⋯ + 𝑛 = ________________________
2. 12 + 22 + 32 + 42 + ⋯ + 𝑛2 = ________________________
3. 13 + 23 + 33 + 43 + ⋯ + 𝑛3 = ________________________
4. 14 + 24 + 34 + 44 + ⋯ + 𝑛4 = ________________________
5. 15 + 25 + 35 + 45 + ⋯ + 𝑛5 = ________________________
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IV.
Series
The sum of the terms of a finite or infinite sequence is called
a(n) _____________.
What you should learn:
How to find sums of infinite
series
Consider the infinite sequence 𝑎1 , 𝑎2 , 𝑎3 , … , 𝑎𝑛 , … The sum of the first 𝑛 terms of the sequence is
called a(n) ________________________ or the ________________________ of the sequence and is
denoted by 𝑎1 + 𝑎2 + 𝑎3 + ⋯ + 𝑎𝑖 + ⋯ = ∑𝑛𝑖=1 𝑎𝑖 . The sum of all terms of the infinite sequence is
called a(n) ________________________ and is denoted by 𝑎1 + 𝑎2 + 𝑎3 + ⋯ + 𝑎𝑖 + ⋯ = ∑∞
𝑖=1 𝑎𝑖 .
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Section 8.1 Examples – Sequences and Series
( 1 ) Find the first five terms of the sequence given by 𝑎𝑛 = 5 + 2𝑛(−1)𝑛 .
( 2 ) Write an expression for the 𝑛th term 𝑎𝑛 of the given sequence.
2, 5, 10, 17, …
( 3 ) A sequence is defined recursively as follows:
𝑎1 = 3,
𝑎𝑘 = 2𝑎𝑘−1 + 1,
Write the first five terms of this sequence.
𝑤ℎ𝑒𝑟𝑒 𝑘 ≥ 2
𝑛!
( 4 ) Evaluate the factorial expression (𝑛+1)!
( 5 ) Find the following sum:
7
∑(2 + 3𝑖)
𝑖=2
( 6 ) For the given series find (a) the third partial sum and (b) the sum.
∞
∑
𝑖=1
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3
10𝑖
Section 8.2 Arithmetic Sequences and Partial Sums
Objective:
In this lesson you learned how to recognize, write, and use arithmetic sequences.
Important Vocabulary
Arithmetic sequence
I.
Finite Arithmetic Sequence
Arithmetic Sequences
The common difference of an arithmetic sequence is:
What you should learn:
How to recognize, write, and
find the 𝑛th terms of
arithmetic sequences
The 𝑛th term of an arithmetic sequence has the form ________________________, where 𝑑 is the
common difference between consecutive terms of the sequence, and 𝑐 = 𝑎1 − 𝑑. An arithmetic
sequence 𝑎𝑛 = 𝑑𝑛 + 𝑐 can be thought of as ________________________, after a shift of ______
units from ______.
The 𝑛th term of an arithmetic sequence has the alternative recursion formula
________________________.
II.
The Sum of a Finite Arithmetic Sequence
The sum of a finite arithmetic sequence with 𝑛 terms is given
by ______________________________________.
What you should learn:
How to find 𝑛th partial sums of
arithmetic sequences
The sum of the first 𝑛 terms of an infinite sequence is called the
___________________________.
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Section 8.2 Examples – Arithmetic Sequences and Partial Sums
( 1 ) Determine whether or not the following sequence is arithmetic. If it is, find the common difference.
7, 3, −1, −5, −9, …
( 2 ) Find a formula for the 𝑛th term of the arithmetic sequence whose common difference is 2 and whose
first term is 7.
( 3 ) Find the sixth term of the arithmetic sequence that begins with 15 and 12.
( 4 ) Find the sum of the first 20 terms of the sequence with 𝑛th term 𝑎𝑛 = 28 − 5𝑛.
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Section 8.3 Geometric Sequences and Series
Objective:
In this lesson you learned how to recognize, write, and use geometric sequences.
Important Vocabulary
Geometric sequence
I.
(Infinite) geometric series
Geometric Sequences
The common ratio of a geometric sequence is:
What you should learn:
How to recognize, write, and
find the 𝑛th terms of
geometric sequences
The 𝑛th term of a geometric sequence has the form ________________________, where 𝑟 is the
common ratio of consecutive terms of the sequence. So, every geometric sequence can be written in
the following form: _________________________________.
If you know the 𝑛th term of a geometric sequence, you can find the (𝑛 + 1)th term by
________________________. That is, 𝑎𝑛+1 =_______.
II.
The Sum of a Finite Geometric Sequence
The sum of a geometric sequence
𝑎1 , 𝑎1 𝑟, 𝑎1 𝑟 2 , 𝑎1 𝑟 3 , 𝑎1 𝑟 4 , … , 𝑎1 𝑟 𝑛−1 with common ratio 𝑟 ≠ 1 is
What you should learn:
How to find 𝑛th partial sums of
geometric sequences
given by ________________________.
When using the formula for the sum of a geometric sequence, be careful to check that the index
begins with 𝑖 = 1. If the index begins at 𝑖 = 0:
III.
Geometric Series
If |𝑟| < 1, then the infinite geometric series 𝑎1 + 𝑎1 𝑟 + 𝑎1 𝑟 2 +
What you should learn:
𝑎1 𝑟 3 + 𝑎1 𝑟 4 + ⋯ + 𝑎1 𝑟 𝑛−1 + ⋯ has the sum
How to find sums of infinite
geometric series
________________________.
If |𝑟| > 1, the series ________________________ a sum.
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Section 8.3 Examples – Geometric Sequences and Series
( 1 ) Determine whether or not the following sequence is geometric. If it is, find the common ratio.
60, 30, 0, −30, −60, …
( 2 ) Write the first five terms of the geometric sequence whose first term is 𝑎1 = 5 and whose common
ratio is −3.
( 3 ) Find the eighth term of the geometric sequence that begins with 15 and 12.
( 4 ) Find the sum:
10
∑ 2(0.5)𝑖
𝑖=1
( 5 ) Find the sum:
12
∑ 4(0.3)𝑖
𝑖=0
( 6 ) If possible, find the sum:
∞
∑ 9(0.25)𝑖−1
𝑖=1
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Section 8.4 The Binomial Theorem
Objective:
In this lesson you learned how to use the Binomial Theorem and Pascal’s Triangle to
calculate binomial coefficients and write binomial expansions.
Important Vocabulary
Binomial coefficients
I.
The Binomial Theorem
Binomial Coefficients
List four general observations about the expansion of (𝑥 + 𝑦)𝑛
for various values of 𝑛.
1.
Pascal’s Triangle
What you should learn:
How to use the Binomial
Theorem to calculate binomial
coefficients
2.
3.
4.
The Binomial Theorem states that in the expansion of (𝑥 + 𝑦)𝑛 = 𝑥 𝑛 + 𝑛𝑥 𝑛−1 𝑦 + ⋯ +
𝑛𝐶𝑟 𝑥
II.
𝑛−𝑟 𝑟
𝑦 + ⋯ + 𝑛𝑥𝑦 𝑛−1 + 𝑦 𝑛 , the coefficient of 𝑥 𝑛−𝑟 𝑦 𝑟 is:
Binomial Expansion
Writing out the coefficients for a binomial that is raised to a
III.
What you should learn:
power is called ______________________.
How to use binomial
coefficients to write binomial
expansions
Pascal’s Triangle
What you should learn:
Construct rows 0 through 6 of Pascal’s Triangle.
How to use Pascal’s Triangle to
calculate binomial coefficients
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Section 8.4 Examples – The Binomial Theorem
( 1 ) Find the binomial coefficient 12𝐶5 .
( 2 ) Write the expansion of the expression (𝑥 + 2)5
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