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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
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multiply by
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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.

\(a_n =\) .