Question

lesson practice a1.1.06

1. match each sequence to its explicit expression.

sequence
a. 4, 10, 16, 22
b. 4, 12, 36, 108
c. 160, 80, 40, 20
d. 320, 320.5, 321, 321.5
explicit expression
320·(1/2)^n
4 + 6(n - 1)
320+1/2(n - 1)
4·3^(n - 1)

1. select all the expressions that could represent the number of tiles in figure n of this pattern.

□ a. 5 + n
□ b. 5n + 1
□ c. 6n - 1
□ d. 6 + 5(n - 1)
□ e. 10n - 4
problems 3 - 4: here is a visual pattern.

1. complete the table with the number of tiles in each figure.
2. write an explicit expression for the number of tiles in figure n.
3. the first four terms of a geometric sequence are: 6, 18, 54, 162. which explicit expression can be used to describe this sequence?

a. 2 + 3n
b. 3^n
c. 2·3^n
d. 6·3^n

1. the first four terms in a sequence are 15, 22, 29, 36. write two expressions for term n of this sequence.

expression 1:
expression 2:

1. a triangular number can be represented as the number of dots in an equilateral triangle.

what is the 8th triangular number?
spiral review
problems 8 - 10: determine the value of...

1. 7x - 8
2. 5(z +...)

problems 11 - 13: decide whether e...
circle one.

1. 25, 5, 1,...
2. 25, 19, 13,...
3. 25, 52, 25, 52,...

reflection

1. put a heart next...
2. use this space...

unit 1 lesson 6

Explanation:

Step1: Identify the type of sequence

This is an arithmetic - sequence with first term \(a_1 = 4\) and common difference \(d=6\). The formula for the \(n\)th term of an arithmetic sequence is \(a_n=a_1+(n - 1)d\).
## Step2: Substitute values
Substitute \(a_1 = 4\) and \(d = 6\) into the formula, we get \(a_n=4+6(n - 1)\).
- # Answer:
\(4+6(n - 1)\)
- **Sequence b: 4, 12, 36, 108**
- # Explanation:
## Step1: Identify the type of sequence
This is a geometric sequence with first term \(a_1 = 4\) and common ratio \(r = 3\). The formula for the \(n\)th term of a geometric sequence is \(a_n=a_1r^{n - 1}\).
## Step2: Substitute values
Substitute \(a_1 = 4\) and \(r = 3\) into the formula, we get \(a_n=4\times3^{n - 1}\).
- # Answer:
\(4\times3^{n - 1}\)
- **Sequence c: 160, 80, 40, 20**
- # Explanation:
## Step1: Identify the type of sequence
This is a geometric sequence with first term \(a_1 = 160\) and common ratio \(r=\frac{1}{2}\). The formula for the \(n\)th term of a geometric sequence is \(a_n=a_1r^{n - 1}\).
## Step2: Substitute values
Substitute \(a_1 = 160\) and \(r=\frac{1}{2}\) into the formula, we get \(a_n=160\times(\frac{1}{2})^{n - 1}=320\times(\frac{1}{2})^{n}\).
- # Answer:
\(320\times(\frac{1}{2})^{n}\)
- **Sequence d: 320, 320.5, 321, 321.5**
- # Explanation:
## Step1: Identify the type of sequence
This is an arithmetic sequence with first term \(a_1 = 320\) and common difference \(d = 0.5=\frac{1}{2}\). The formula for the \(n\)th term of an arithmetic sequence is \(a_n=a_1+(n - 1)d\).
## Step2: Substitute values
Substitute \(a_1 = 320\) and \(d=\frac{1}{2}\) into the formula, we get \(a_n=320+\frac{1}{2}(n - 1)\).
- # Answer:
\(320+\frac{1}{2}(n - 1)\)

2. Select all the expressions that could represent the number of tiles in Figure \(n\) of this pattern

- # Explanation:
## Step1: Assume the first - term and common - difference for an arithmetic - sequence approach
Let's assume the number of tiles forms an arithmetic sequence. If we check the general form of an arithmetic sequence \(a_n=a_1+(n - 1)d\).
## Step2: Analyze each option
For option A: \(a_n=5 + n\), when \(n = 1\), \(a_1=6\).
For option B: \(a_n=5n+1\), when \(n = 1\), \(a_1=6\).
For option C: \(a_n=6n - 1\), when \(n = 1\), \(a_1=5\).
For option D: \(a_n=6+5(n - 1)=6 + 5n-5=5n + 1\).
For option E: \(a_n=10n - 4\), when \(n = 1\), \(a_1=6\).
If we assume the first - term \(a_1 = 6\) and common difference \(d = 5\), the correct expressions are \(5n + 1\) and \(6+5(n - 1)\).
- # Answer:
B. \(5n + 1\), D. \(6+5(n - 1)\)

3. Complete the table with the number of tiles in each figure (assuming a visual pattern not shown completely here, but we can use the arithmetic - sequence approach)

Let's assume the number of tiles in the figures forms an arithmetic sequence. If the first - figure has \(a_1\) tiles and common difference \(d\).
| Figure | # of Tiles |
|---|---|
| 1 | \(a_1\) (assume \(a_1\) is known from the pattern) |
| 2 | \(a_1 + d\) |
| 3 | \(a_1+2d\) |
| 4 | \(a_1 + 3d\) |
| 15 | \(a_1+14d\) |

4. Write an explicit expression for the number of tiles in Figure \(n\)

- # Explanation:
## Step1: Use the arithmetic - sequence formula
The formula for the \(n\)th term of an arithmetic sequence is \(a_n=a_1+(n - 1)d\), where \(a_1\) is the number of tiles in the first figure and \(d\) is the common difference in the number of tiles between consecutive figures.
- # Answer:
\(a_n=a_1+(n - 1)d\)

5. The first four terms of a geometric sequence are: 6, 18, 54, 162. Which explicit expression can be used to describe this sequence?

- # Explanation:
## Step1: Identify the first term and common ratio
The first term \(a_1 = 6\) and the common ratio \(r=\frac{18}{6}=3\).
## Step2: Use the geometric - sequence formula
The formula for the \(n\)th term of a geometric sequence is \(a_n=a_1r^{n - 1}\). Substitute \(a_1 = 6\) and \(r = 3\) into the formula: \(a_n=6\times3^{n - 1}=2\times3\times3^{n - 1}=2\times3^{n}\).
- # Answer:
C. \(2\times3^{n}\)

Answer:

1. Match each sequence to its explicit expression

• Sequence a: 4, 10, 16, 22
• # Explanation:

Step1: Identify the type of sequence

This is an arithmetic - sequence with first term \(a_1 = 4\) and common difference \(d=6\). The formula for the \(n\)th term of an arithmetic sequence is \(a_n=a_1+(n - 1)d\).

Step2: Substitute values

Substitute \(a_1 = 4\) and \(d = 6\) into the formula, we get \(a_n=4+6(n - 1)\).

• # Answer:

\(4+6(n - 1)\)

• Sequence b: 4, 12, 36, 108
• # Explanation:

Step1: Identify the type of sequence

This is a geometric sequence with first term \(a_1 = 4\) and common ratio \(r = 3\). The formula for the \(n\)th term of a geometric sequence is \(a_n=a_1r^{n - 1}\).

Step2: Substitute values

Substitute \(a_1 = 4\) and \(r = 3\) into the formula, we get \(a_n=4\times3^{n - 1}\).

• # Answer:

\(4\times3^{n - 1}\)

• Sequence c: 160, 80, 40, 20
• # Explanation:

Step1: Identify the type of sequence

This is a geometric sequence with first term \(a_1 = 160\) and common ratio \(r=\frac{1}{2}\). The formula for the \(n\)th term of a geometric sequence is \(a_n=a_1r^{n - 1}\).

Step2: Substitute values

Substitute \(a_1 = 160\) and \(r=\frac{1}{2}\) into the formula, we get \(a_n=160\times(\frac{1}{2})^{n - 1}=320\times(\frac{1}{2})^{n}\).

• # Answer:

\(320\times(\frac{1}{2})^{n}\)

• Sequence d: 320, 320.5, 321, 321.5
• # Explanation:

Step1: Identify the type of sequence

This is an arithmetic sequence with first term \(a_1 = 320\) and common difference \(d = 0.5=\frac{1}{2}\). The formula for the \(n\)th term of an arithmetic sequence is \(a_n=a_1+(n - 1)d\).

Step2: Substitute values

Substitute \(a_1 = 320\) and \(d=\frac{1}{2}\) into the formula, we get \(a_n=320+\frac{1}{2}(n - 1)\).

• # Answer:

\(320+\frac{1}{2}(n - 1)\)

2. Select all the expressions that could represent the number of tiles in Figure \(n\) of this pattern

• # Explanation:

Step1: Assume the first - term and common - difference for an arithmetic - sequence approach

Let's assume the number of tiles forms an arithmetic sequence. If we check the general form of an arithmetic sequence \(a_n=a_1+(n - 1)d\).

Step2: Analyze each option

For option A: \(a_n=5 + n\), when \(n = 1\), \(a_1=6\).
For option B: \(a_n=5n+1\), when \(n = 1\), \(a_1=6\).
For option C: \(a_n=6n - 1\), when \(n = 1\), \(a_1=5\).
For option D: \(a_n=6+5(n - 1)=6 + 5n-5=5n + 1\).
For option E: \(a_n=10n - 4\), when \(n = 1\), \(a_1=6\).
If we assume the first - term \(a_1 = 6\) and common difference \(d = 5\), the correct expressions are \(5n + 1\) and \(6+5(n - 1)\).

• # Answer:

B. \(5n + 1\), D. \(6+5(n - 1)\)

3. Complete the table with the number of tiles in each figure (assuming a visual pattern not shown completely here, but we can use the arithmetic - sequence approach)

Let's assume the number of tiles in the figures forms an arithmetic sequence. If the first - figure has \(a_1\) tiles and common difference \(d\).

Figure	# of Tiles
2	\(a_1 + d\)
3	\(a_1+2d\)
4	\(a_1 + 3d\)
15	\(a_1+14d\)

4. Write an explicit expression for the number of tiles in Figure \(n\)

• # Explanation:

Step1: Use the arithmetic - sequence formula

The formula for the \(n\)th term of an arithmetic sequence is \(a_n=a_1+(n - 1)d\), where \(a_1\) is the number of tiles in the first figure and \(d\) is the common difference in the number of tiles between consecutive figures.

• # Answer:

\(a_n=a_1+(n - 1)d\)

5. The first four terms of a geometric sequence are: 6, 18, 54, 162. Which explicit expression can be used to describe this sequence?

• # Explanation:

Step1: Identify the first term and common ratio

The first term \(a_1 = 6\) and the common ratio \(r=\frac{18}{6}=3\).

Step2: Use the geometric - sequence formula

The formula for the \(n\)th term of a geometric sequence is \(a_n=a_1r^{n - 1}\). Substitute \(a_1 = 6\) and \(r = 3\) into the formula: \(a_n=6\times3^{n - 1}=2\times3\times3^{n - 1}=2\times3^{n}\).

• # Answer:

C. \(2\times3^{n}\)