Download Sequences and Series

Knight’s Charge
8/26/15
Solve:
1) 11 − 3𝑥 = −2
2) 4𝑥 2 + 8𝑥 = 5
3) 𝑥 2 − 2𝑥 = −4
Review
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Glencoe Precalculus Book
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Check Homework
Set D Practice Wkst
8/26/15
Sequences and Series
Unit 1
Consider this:

A pyramid of logs has 2 logs in the top row, 4 logs
in the second row from the top, 6 logs in the third
row from the top, and so on, until there are 200
logs in the bottom row.
Intro to Sequences
A pyramid of logs has 2 logs in the top row, 4 logs in the second
row from the top, 6 logs in the third row from the top, and so
on, until there are 200 logs in the bottom row.

Write and interpret the first 10 terms of the sequence of
numbers generated from the example.

Identify the pattern in the sequence of numbers.

Write the formula for the nth term of the sequence and use
it to find the number of logs in, say, the 76th row

Compute the number of logs in the first 12 rows combined.

What is the total number of logs in the pyramid?
Intro to Sequences
Intro to Arithmetic Series:
One of the most famous legends in the lore of mathematics
concerns German mathematician Carl Friedrich Gauss.
 One version of the story has it that in primary school
after the young Gauss misbehaved, his teacher, J.G.
Büttner, gave him a task: add the numbers from 1 to100.
The young Gauss reputedly produced the correct answer
within seconds, to the astonishment of his teacher and his
assistant, Martin Bartels. Can you?
 Gauss's realized that pairwise addition of terms from
opposite ends of the list yielded identical intermediate
sums: 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, and so
on, for a total sum of 50 × 101 = 5050.
Arithmetic Series
Notation
Consider the sequence: 1, 4, 7, 10, 13, …


𝑛 = the term number (think of it as the term’s place in line)
𝑎𝑛 = the nth term




𝑎1 represents the FIRST term.
𝑎2 represents the SECOND term.
𝑎3 represents the THIRD term.
𝑎4 represents the FOURTH term, etc…

𝑎𝑛−1 = the previous term
𝑎𝑛+1 = the next term

IMPLICIT FORMULA: requires knowing the previous term

EXPLICIT FORMULA: requires only knowing the desired n.

General Sequences
Fill in the chart.
SEQUENCE
IMPLICIT
FORMULA
EXPLICIT
FORMULA
1, 2, 4, 8, 16, …
𝑎1 = 10
𝑎𝑛 = 5 + 𝑎𝑛−1
𝑎𝑛 = 𝑛2
General Sequences
Find the first six terms for each sequence:

𝑎𝑘 = 3𝑘

𝑑𝑘 = 2𝑑𝑘−1 , 𝑑1 = 5

𝑏𝑘 = 𝑘 + 5

𝑒𝑘 = 𝑒𝑘−1 2 , 𝑒1 = 2

𝑐𝑘 = 𝑘 2 − 3𝑘

𝑓𝑘 = 𝑓𝑘−1 + 4, 𝑓1 = 3
Arithmetic, Geometric, or Neither?
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
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An arithmetic sequence is one where a constant value is
added to each term to get the next term.
example: {5, 7, 9, 11, …}
A geometric sequence is one where a constant value is
multiplied by each term to get the next term.
example: {5, 10, 20, 40, …}
EXAMPLE: Determine whether each of the following
sequences is arithmetic, geometric, or neither:
a.
1 1 1 1
, , , ,
2 4 8 16
...
GEOMETRIC
b. {9, -1, -11, -21, ...} ARITHMETIC
c. {0, 1, 1, 2, 3, 5, 8, 13, 21,...} NEITHER
Fibonacci Sequence
Formal Definition of an Arithmetic Sequence

A sequence is arithmetic if there exists a number d, called
the common difference, such that for 𝑑 = 𝑎𝑛 − 𝑎𝑛−1 for
𝑛 ≥ 2.

In other words, if we start with a particular first term, and
then add the same number successively, we obtain an
arithmetic sequence.
Arithmetic Sequences
Example: Write an explicit formula for the
sequence {10, 15, 20, 25, …}.

Note: this sequence is arithmetic with a common difference (d) of 5.

Make a table of values for the terms of the sequence.
Then graph the table.
What do you notice about
the graph?
It’s LINEAR……
Can you write the equation of
the line/sequence now?
Yes, the equation of the line is 𝑦 = 5𝑥 + 5…
So the formula for the sequence is
Arithmetic Sequences
𝑎𝑛 = 5𝑛 + 5.
Example: Write an explicit formula for the
sequence {10, 15, 20, 25, …}.

So how could we write the formula WITHOUT having to
graph it?

In general, the explicit formula for an arithmetic sequence is
given by 𝑎𝑛 = 𝑎1 + 𝑑(𝑛 − 1).
Arithmetic Sequences
Example: Fill in the chart for each arithmetic
sequence shown.
SEQUENCE
IMPLICIT
FORMULA
EXPLICIT
FORMULA
2, 5, 8, 11, 14, …
𝑎𝑛 = 4𝑛 − 3
𝑎1 = −50
𝑎𝑛 = 10 + 𝑎𝑛−1
Arithmetic Sequences
100th term
Example: Given 𝑎1 = 5 and 𝑎10 = −22, find
the 100th term of the sequence.
Arithmetic Sequences
Example: Given 𝑎6 = 20 and 𝑎10 = 32, find
the 25th term of the sequence.
Arithmetic Sequences
Arithmetic Means

Example: Form an arithmetic sequence that has 3
arithmetic means between 15 and 35.

Example: Form an arithmetic sequence that has 4
arithmetic means between 13 and 15.
Arithmetic SERIES

What is an arithmetic SERIES? --the SUM of an
indicated number of terms of a sequence.
Arithmetic Sequence:
1, 3, 5, 7, 9, …
Arithmetic Series:
1+3+5+7+9+⋯
Sum of a FINITE Arithmetic Sequence

The sum of a finite arithmetic sequence with common
𝑛
difference d is 𝑆𝑛 = 𝑎1 + 𝑎𝑛 .
2

Example: Find the sum of the first 15 terms of the
sequence {1, 5, 9, 13, … }.

Example: Find the sum of the first 100 terms of the
sequence {-18, -13, -8, -3, 2,…}.
Arithmetic Series
Example: Given the sum of the first 20 terms
of a sequence that starts with 5 is 220, find
the 20th term.
Arithmetic Series
Example: Given the sum of the first 15
terms of an arithmetic sequence is 165 and
the first term is −3, find…

the common difference.

the 15th term.

the explicit formula for the sequence.

the sum of the first 20 terms of the sequence.
Arithmetic Series
Application of Arithmetic Series
A corner section of a stadium has 14 seats along the front
row and adds one more seat to each successive row. If the
top row has 35 seats, how many seats are in that section?
Arithmetic Series
Homework

Pre-precal review Set J Extra Practice

Textbook p. 605 #1-25 Odd
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Arithmetic Sequences