Arithmetic Sequences: Dot Patterns and Formula Application

Q1. Complete the table to show the number of dots in designs 1 to 4.

Background

Topic: Arithmetic Sequences

This question is testing your ability to recognize and extend a pattern in a sequence, specifically an arithmetic sequence where each term increases by a constant amount.

Key Terms and Formulas:

• Arithmetic sequence: A sequence of numbers where the difference between consecutive terms is constant.

• First term ($a_1$): The starting value of the sequence.

• Common difference ($d$): The amount added to each term to get the next term.

• General formula for the $n$th term:

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

Step-by-Step Guidance

1. Observe the dot patterns for each design and count the number of dots in each.

2. Fill in the table for designs 1 to 4 based on your counts. Notice how the number of dots increases from one design to the next.

3. Identify the first term ($a_1$) and the common difference ($d$) by looking at the table. The first term is the number of dots in design 1, and the common difference is the increase in dots from one design to the next.

4. Check your table to ensure the pattern is consistent: each new design should have the previous number of dots plus the common difference.

Try solving on your own before revealing the answer!

Q2. Write the first term and the common difference.

Background

Topic: Identifying Sequence Parameters

This question is testing your ability to identify the starting value and the constant difference in an arithmetic sequence.

Key Terms:

• First term ($a_1$): The value of the sequence at design 1.

• Common difference ($d$): The difference between each consecutive term.

Step-by-Step Guidance

1. Look at the table you completed in part (a) and identify the first term ($a_1$).

2. Calculate the common difference ($d$) by subtracting the number of dots in design 1 from design 2, or design 2 from design 3, etc.

3. Check that the difference is consistent for all consecutive designs.

Try solving on your own before revealing the answer!

Q3. How many dots are in design 8?

Background

Topic: Applying the Arithmetic Sequence Formula

This question is testing your ability to use the arithmetic sequence formula to find the number of dots in a specific design number.

Key Formula:

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

• $a_n$ = number of dots in design $n$

• $a_1$ = first term (number of dots in design 1)

• $d$ = common difference

• $n$ = design number

Step-by-Step Guidance

1. Identify the values for $a_1$, $d$, and $n$ (for design 8).

2. Set up the formula: $a_8 = a_1 + (8-1)d$

3. Calculate $(8-1)$ to find how many times the common difference is added to the first term.

4. Multiply the common difference by $(8-1)$ and add it to the first term.

Try solving on your own before revealing the answer!

Final Answer: 23 dots

$a_8 = 2 + (8-1) \times 3 = 2 + 21 = 23$

We used the arithmetic sequence formula and substituted the correct values for the first term and common difference.