Question

1. jogging roberta jogs 3 laps the first day, 5 laps the second day, 7 laps the third day, and so on. on which day will roberta jog 13 laps if the pattern continues?

describe the relationship between the terms in each arithmetic sequence. then write the next three terms in each sequence.

1. 2, 9, 16, 23, …
2. 50, 53, 56, 59, …
3. 0.2, 0.6, 1.0, 1.4, …
4. 81, 90, 99, 108, …

Explanation:

Question 9

Step 1: Identify the sequence type

The number of laps Roberta jogs each day forms an arithmetic sequence where the first term \(a_1 = 3\) and the common difference \(d=2\) (since \(5 - 3=2\), \(7 - 5 = 2\), etc.). The formula for the \(n\)-th term of an arithmetic sequence is \(a_n=a_1+(n - 1)d\).

Step 2: Substitute values and solve for \(n\)

We want to find \(n\) when \(a_n = 13\). Substitute \(a_1 = 3\), \(d = 2\) and \(a_n=13\) into the formula:
\[13=3+(n - 1)\times2\]

Step 3: Simplify the equation

Subtract 3 from both sides:
\[13 - 3=(n - 1)\times2\]
\[10=(n - 1)\times2\]

Step 4: Solve for \(n\)

Divide both sides by 2:
\[n - 1=\frac{10}{2}=5\]
Add 1 to both sides:
\[n=5 + 1=6\]

Step 1: Find the common difference

To find the common difference \(d\), subtract consecutive terms. \(9-2 = 7\), \(16 - 9=7\), \(23-16 = 7\). So the common difference \(d = 7\). The relationship is that each term is 7 more than the previous term.

Step 2: Find the next three terms

To find the next term after 23, add 7: \(23+7 = 30\).
Then the next term: \(30+7=37\).
Then the next term: \(37 + 7=44\).

Step 1: Find the common difference

Subtract consecutive terms: \(53 - 50=3\), \(56 - 53 = 3\), \(59 - 56=3\). So \(d = 3\). The relationship is each term is 3 more than the previous term.

Step 2: Find the next three terms

Next term after 59: \(59+3 = 62\).
Next term: \(62+3 = 65\).
Next term: \(65+3 = 68\).

Answer:

6

Question 10