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Discrete Mathematics: An Open Introduction, 4th edition

Oscar Levin

Worksheet Preview Activity
A4US

1.
Explore the first sequence of dots from the Investigate! activity. We will let \(a_n\) represent the number of dots in figure \(n\text{.}\) The sequence starts \(1, 5, 9,\ldots\)
(a)
How many dots would you expect in the next two figures in the sequence?
Dots in \(n = 3\) figure: \(a_3 =\).
Dots in \(n = 4\) figure: \(a_4 =\).
(b)
How is the sequence growing? To get the next figure, take the current figure and
  • add
  • multiply by
the constant dots.
(c)
Let \(a_n\) be the number of dots in figure \(n\text{.}\) Write a recursive definition for \(a_n\text{.}\)
\(a_n =\) ; with \(a_0 =\) .
(d)
Guess a closed formula for the number of dots in the \(n\)th figure.
\(a_n =\) .
2.
Now look at the second sequence of dots from the Investigate! activity. We will let \(a_n\) represent the number of dots in figure \(n\text{.}\) The sequence starts \(2, 6, 18,\ldots\)
(a)
How many dots would you expect in the next two figures in the sequence?
Dots in \(n = 3\) figure: \(a_3 =\).
Dots in \(n = 4\) figure: \(a_4 =\).
(b)
How is the sequence growing? To get the next figure, take the current figure and
  • add
  • multiply by
the constant .
(c)
Let \(a_n\) be the number of dots in figure \(n\text{.}\) Write a recursive definition for \(a_n\text{.}\)
\(a_n =\) ; with \(a_0 =\) .
(d)
Guess a closed formula for the number of dots in the \(n\)th figure.
\(a_n =\) .
3.
Now look at the third sequence of dots from the Investigate! activity. We will let \(a_n\) represent the number of dots in figure \(n\text{.}\) The sequence starts \(1, 3, 6, 10,\ldots\)
(a)
How many dots would you expect in the next two figures in the sequence?
Dots in \(n = 5\) figure: \(a_5 =\).
Dots in \(n = 6\) figure: \(a_6 =\).
(b)
Let \(a_n\) be the number of dots in figure \(n\text{.}\) Write a recursive definition for \(a_n\text{.}\)
\(a_n =\) ; with \(a_1 =\) .
(c)
Guess a closed formula for the number of dots in the \(n\)th figure.
\(a_n =\) .
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