Linear Equations and Arithmetic Sequences

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Linear Equations and Arithmetic Sequences

• Lesson 3.1

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• You can solve many rate problems by using
  recursion.
• Matias wants to call his aunt in Chile on her
  birthday. He learned that placing the call costs
  2.27 and that each minute he talks costs 1.37.
  How much would it cost to talk for 30 minutes?
  You can calculate the cost of Matiass phone call
  with the recursive formula

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To find the cost of a 30-minute phone call,
calculate the first 30 terms, as shown in the
calculator screen.
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As you learned in algebra, you or Matias can also
find the cost of a 30-minute call by using the
linear equation y 2.27 1.37x where x is the
length of the phone call in minutes and y is the
cost in dollars.
If the phone company always rounds up the length
of the call to the nearest whole minute, then the
costs become a sequence of discrete points, and
you can write the relationship as an explicit
formula, un 2.27 1.37n, where n is the length
of the call in whole minutes and un is the cost
in dollars.
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Example A

• Consider the recursively defined arithmetic
  sequence
• Find an explicit formula for the sequence.

Notice you start with 2 and keep adding a common
difference or rate of change is 6.
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In general, when you write the formula for a
sequence, you use n to represent the number of
the term and un to represent the term itself.
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Example A

• Consider the recursively defined arithmetic
  sequence
• Use the explicit formula to find u22 .

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Example A

• Consider the recursively defined arithmetic
  sequence
• Find the value of n so that un 86.

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You graphed sequences of points (n, un ) in
Chapter 1. The term number n is a whole number
0, 1, 2, 3, . . . . So, using different values
for n will produce a set of discrete points. The
points on this graph show the arithmetic sequence
from Example A.
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When n increases by 1, un increases by 6, the
common difference. The change in the y-value is
6 that corresponds to a change of 1 in the
x-value. So the points representing the sequence
lie on a line with a slope of 6.
In general, the common difference, or rate of
change, between consecutive terms of an
arithmetic sequence is the slope of the line
through those points.
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The pair (0, 2) names the starting value 2, which
is the y-intercept. Using the intercept form of a
linear equation, you can now write an equation of
the line through the points of the sequence as y
26x, or y6x 2.
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Match Point

• Below are three recursive formulas, three graphs,
  and three linear equations.

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• Match the recursive formulas, graphs, and linear
  equations that go together. (Not all of the
  appropriate matches are listed. If the recursive
  rule, graph, or equation is missing, you will
  need to create it.)

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• Write a brief statement relating the starting
  value and common difference of an arithmetic
  sequence to the corresponding equation yabx.

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• Are points (n, un) of an arithmetic sequence
  always collinear? Write a brief statement
  supporting your answer.

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Example B

• Retta typically spends 2 a day on lunch. She
  notices that she has 17 left after todays
  lunch. She thinks of this sequence to model her
  daily cash balance.
• Find the explicit formula that represents her
  daily cash balance and an equation of the line
  through the points of this sequence.

Each term is 2 less than the previous term, so
the common difference of the arithmetic sequence
and the slope of the line are both -2. The term
t1 is 17, so the previous term, t0, or the
y-intercept, is 19. y 19 - 2x.
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Example B

• Retta typically spends 2 a day on lunch. She
  notices that she has 17 left after todays
  lunch. She thinks of this sequence to model her
  daily cash balance.
• How useful is this formula for predicting how
  much money Retta will have each day?

We dont know whether Retta has any other
expenses she might encounter. The formula could
be valid for eight more days, until she has 1
left (on t9), as long as she gets no more money
and spends only 2 per day.
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