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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 .

The sequence starts

1, 5, 9, \dots

🔗

(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 .

Write a recursive definition for

a_{n} .

🔗

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 .

The sequence starts

2, 6, 18, \dots

🔗

(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 .

Write a recursive definition for

a_{n} .

🔗

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 .

The sequence starts

1, 3, 6, 10, \dots

🔗

(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 .

Write a recursive definition for

a_{n} .

🔗

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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1.

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(d)

2.

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(b)

(c)

(d)

3.

(a)

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(c)

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