Skip to main content
Logo image

Discrete Mathematics An Open Introduction, 4th Edition

Oscar Levin

Print preview

Paper size

Worksheet Preview Activity

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: \(a_4 =\).
🔗
Answer 1.
\(13\)
🔗
🔗
Answer 2.
\(17\)
🔗
🔗
🔗
(b)
How is the sequence growing? To get the next figure, take the current figure and
the constant dots.
🔗
Answer 1.
\(\text{add}\)
🔗
🔗
Answer 2.
\(4\)
🔗
🔗
🔗
(c)
Let \(a_n\) be the number of dots in figure \(n\text{.}\) Write a recursive definition for \(a_n\text{.}\)
🔗
\(a_n =\) ; with \(a_0 =\) .
🔗
Answer 1.
\(a\mathopen{}\left(n-1\right)+4\)
🔗
🔗
Answer 2.
\(1\)
🔗
🔗
🔗
(d)
Guess a closed formula for the number of dots in the \(n\)th figure.
🔗
\(a_n =\) .
🔗
Answer.
\(1+4n\)
🔗
🔗
🔗
🔗
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: \(a_4 =\).
🔗
Answer 1.
\(54\)
🔗
🔗
Answer 2.
\(162\)
🔗
🔗
🔗
(b)
How is the sequence growing? To get the next figure, take the current figure and
the constant .
🔗
Answer 1.
\(\text{multiply by}\)
🔗
🔗
Answer 2.
\(3\)
🔗
🔗
🔗
(c)
Let \(a_n\) be the number of dots in figure \(n\text{.}\) Write a recursive definition for \(a_n\text{.}\)
🔗
\(a_n =\) ; with \(a_0 =\) .
🔗
Answer 1.
\(3a\mathopen{}\left(n-1\right)\)
🔗
🔗
Answer 2.
\(2\)
🔗
🔗
🔗
(d)
Guess a closed formula for the number of dots in the \(n\)th figure.
🔗
\(a_n =\) .
🔗
Answer.
\(2\cdot 3^{n}\)
🔗
🔗
🔗
🔗
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: \(a_6 =\).
🔗
Answer 1.
\(15\)
🔗
🔗
Answer 2.
\(21\)
🔗
🔗
🔗
(b)
Let \(a_n\) be the number of dots in figure \(n\text{.}\) Write a recursive definition for \(a_n\text{.}\)
🔗
\(a_n =\) ; with \(a_1 =\) .
🔗
Answer 1.
\(a\mathopen{}\left(n-1\right)+n\)
🔗
🔗
Answer 2.
\(1\)
🔗
🔗
🔗
(c)
Guess a closed formula for the number of dots in the \(n\)th figure.
🔗
\(a_n =\) .
🔗
Answer.
\(\frac{n\mathopen{}\left(n+1\right)}{2}\)
🔗
🔗
🔗
🔗
🔗
PrevTopNext
Feedback