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# 9.5: Lines of Numbers

Difficulty Level: At Grade Created by: CK-12

Look at the pattern below. How would you describe it? Can you answer the questions about the pattern? In this concept, we will practice working with number patterns.

### Guidance

In order to examine patterns and answer questions about patterns like the ones above, we can use the problem solving steps to help.

• First, describe what you see in the pattern. What numbers are there? When does it start repeating?
• Second, identify what your job is. In these problems, your job will be to answer the questions.
• Third, make a plan. In these problems, your plan should be to figure out how many times the pattern repeats before reaching the given number.
• Fourth, solve. Answer the questions.

#### Example A

Look at the pattern below and answer the questions.

Solution:

We can use problem solving steps to help us answer the questions about the pattern.

Describe: The number pattern keeps repeating the numbers 1, 2 3, 4.

My Job: Figure out the 25th and 75th numbers and the sum of the first 75 numbers.

Plan: The numbers are in sets of four: 1, 2, 3, and 4. Find how many sets of four numbers there are and how many numbers are left over to find the 25th and 75th numbers. To find the sum of the first 75 numbers, multiply the number of sets of four numbers by the sum of the four numbers. Then add in the left over numbers.

Solve: a. \begin{align*}25 \div 4\end{align*} is 6 with 1 left over and \begin{align*}4 \times 6 =24\end{align*}. So, the 24th number will be 4 (the last number in the set). Therefore, the 25th number will be 1.

b. \begin{align*}75 \div 4\end{align*} is 18 with 3 left over and \begin{align*}4 \times 18=72\end{align*}. So, the 72nd number will be 4, the 73rd number will be 1, the 74th number will be 2, and the 75th number will be 3.
c. 10 is the sum of one set of 1, 2, 3, 4. There are 18 sets of four numbers ending with 72. \begin{align*}10 \times 18=180\end{align*} is the sum of the first 72 numbers. \begin{align*}1+2+3=6\end{align*} is the sum of the three left over numbers. So, \begin{align*}180+6=186\end{align*} is the sum of the first 75 numbers in the pattern.

Check: a. The 1st, 5th, 9th, and so forth numbers are 1s. All positions one more than a multiple of 4 are 1s. So the 25th number is 1.

b. 73 is one more than a multiple of 4, so the 73rd number is 1, the 74th number is 2, and the 75th number is 3.
c. There are 18 sets of 4 in 75, with three left over. The sum of the first 18 sets of four numbers is \begin{align*}10\times 18=180\end{align*} and the sum of the three extra numbers is 6, so the total sum is 186.

#### Example B

Look at the pattern below and answer the questions.

Solution:

We can use problem solving steps to help us answer the questions about the pattern.

Describe: The number pattern keeps repeating the numbers 2, 3, 4.

My Job: Figure out the 40th and 110th numbers and the sum of the first 110 numbers.

Plan: The numbers are in sets of three: 2, 3, and 4. Find how many sets of three numbers there are and how many numbers are left over to find the 40th and 110th numbers. To find the sum of the first 110 numbers, multiply the number of sets of three numbers by the sum of the three numbers. Then add in the left over numbers.

Solve: a. \begin{align*}40 \div 3\end{align*} is 13 with 1 left over and \begin{align*}3 \times 13 =39\end{align*}. So, the 39th number will be 4 (the last number in the set). Therefore, the 40th number will be 2.

b. \begin{align*}110 \div 3\end{align*} is 36 with 2 left over and \begin{align*}3 \times 36=108\end{align*}. So, the 108th number will be 4. The 109th number will be 2, and the 110th number will be 3.
c. 9 is the sum of one set of 2, 3, 4. There are 36 sets of three numbers ending with 108. \begin{align*}9 \times 36=324\end{align*} is the sum of the first 108 numbers. \begin{align*}2+3=5\end{align*} is the sum of the two left over numbers. So, \begin{align*}324+5=329\end{align*} is the sum of the first 110 numbers in the pattern.

Check: a. The 1st, 4th, 7th, and so forth numbers are 2s. All positions one more than a multiple of 3 are 2s. So the 40th number is 2.

b. 109 is one more than a multiple of 3, so the 109th number is 2 and the 110th number is 3.
c. There are 36 sets of 3 in 110, with two left over. The sum of the first 36 sets of three numbers is \begin{align*}9\times 36=324\end{align*} and the sum of the two extra numbers is 5, so the total sum is 329.

#### Example C

Look at the pattern below and answer the questions.

Solution:

We can use problem solving steps to help us answer the questions about the pattern.

Describe: The number pattern keeps repeating the numbers 1, 3, 5, 7.

My Job: Figure out the 70th and 175th numbers and the sum of the first 175 numbers.

Plan: The numbers are in sets of four: 1, 3, 5, and 7. Find how many sets of four numbers there are and how many numbers are left over to find the 70th and 175th numbers. To find the sum of the first 175 numbers, multiply the number of sets of four numbers by the sum of the four numbers. Then add in the left over numbers.

Solve: a. \begin{align*}70 \div 4\end{align*} is 17 with 2 left over and \begin{align*}4 \times 17 =68\end{align*}. So, the 68th number will be 7 (the last number in the set). Therefore, the 69th number will be 1, and the 70th number will be 3.

b. \begin{align*}175 \div 4\end{align*} is 43 with 3 left over and \begin{align*}4 \times 43=172\end{align*}. So, the 172nd number will be 7, the 173rd number will be 1, the 174th number will be 3, and the 175th number will be 5.
c. 16 is the sum of one set of 1, 3, 5, 7. There are 43 sets of four numbers ending with 172. \begin{align*}16 \times 43=688\end{align*} is the sum of the first 172 numbers. \begin{align*}1+3+5=9\end{align*} is the sum of the three left over numbers. So, \begin{align*}688+9=697\end{align*} is the sum of the first 175 numbers in the pattern.

Check: a. The 1st, 5th, 9th, and so forth numbers are 1s. All positions one more than a multiple of 4 are 1s. So the 69th number is 1 and the 70th number is 3.

b. 173 is one more than a multiple of 4, so the 173rd number is 1, the 174th number is 3, and the 175th number is 5.
c. There are 43 sets of 4 in 175, with three left over. The sum of the first 43 sets of four numbers is \begin{align*}16\times 43=688\end{align*} and the sum of the three extra numbers is 9, so the total sum is 697.

#### Concept Problem Revisited

We can use problem solving steps to help us answer the questions about the pattern.

Describe: The number pattern keeps repeating the numbers 3, 4, and 4.

My Job: Figure out the 50th and 200th numbers and the sum of the first 200 numbers.

Plan: The numbers are in sets of three: 3, 4, and 4. Find how many sets of three numbers there are and how many numbers are left over to find the 50th and 200th numbers. To find the sum of the first 200 numbers, multiply the number of sets of three numbers by the sum of the three numbers. Then add in the left over numbers.

Solve: a. \begin{align*}50 \div 3\end{align*} is 16 with 2 left over and \begin{align*}3 \times 16 =48\end{align*}. So, the 48th number will be 4 (the last number in the set). Therefore, the 49th number will be 3, and the 50th number will be 4.

b. \begin{align*}200 \div 3\end{align*} is 66 with 2 left over and \begin{align*}3 \times 66=198\end{align*}. So, the 198th number will be 4. The 199th number will be 3, and the 200th number will be 4.
c. 11 is the sum of one set of 3, 4, 4. There are 66 sets of three numbers ending with 198. \begin{align*}11 \times 66=726\end{align*} is the sum of the first 198 numbers. \begin{align*}3+4=7\end{align*} is the sum of the two left over numbers. So, \begin{align*}726+7=733\end{align*} is the sum of the first 200 numbers in the pattern.

Check: a. The 1st, 4th, 7th, and so forth numbers are 3s. All positions one more than a multiple of 3 are 3s. So the 49th number is 3 and the 50th number is 4.

b. 199 is one more than a multiple of 3, so the 199th number is 3 and the 200th number is 4.
c. There are 66 sets of 3 in 200, with two left over. The sum of the first 66 sets of three numbers is \begin{align*}11\times 6=726\end{align*} and the sum of the two extra numbers is 7, so the total sum is 733.

### Vocabulary

One type of pattern is when a set of objects repeats over and over. In this concept, we saw patterns of numbers where sets of numbers repeated to form a pattern. With any pattern you should be able to describe the pattern and how to get from one step of the pattern to the next.

### Guided Practice

Look at each pattern and answer the questions.

1.

2.

3.

1. a. The 47th number is 8. \begin{align*}47\div 3\end{align*} is 15 with two left over. So, the 45th number is 9, the 46th number is 7, and the 47th number is 8.

b. The 100th number is 7. \begin{align*}100 \div 3 \end{align*} is 33 with 1 left over. So, the 99th number is 9 and the 100th number is 7.
c. The sum of the first 100 numbers is 809. The sum of one set of 7, 8, 9 is 24. From the answer to question b, we know that 33 sets of three numbers ends with 99. The sum of the first 99 numbers is \begin{align*}33\times 24=802\end{align*}. The 100th number is 7, so the sum of the first 100 numbers in the pattern is \begin{align*}802+7=809\end{align*}.

2. a. The 27th number is 6. \begin{align*}27 \div 4\end{align*} is 6 with three left over. So, the 24th number is 8, the 25th number is 4, the 26th number is 4, and the 27th number is 6.

b. The 78th number is 4. \begin{align*}78 \div 4\end{align*} is 19 with 2 left over. So, the 76th number is 8, the 77th number is 4, and the 78th number is 4.
c. The sum of the first 78 numbers is 426. The sum of one set of 4, 4, 6, and 8 is 22. From the answer to question b, we know that 19 sets of four numbers ends with 76. The sum of the first 76 numbers is \begin{align*}19\times 22=418\end{align*}. The two left over numbers are 4 and 4, and their sum is 8. So the sum of the first 78 numbers in the pattern is \begin{align*}418+8=426\end{align*}.

3. a. The 38th number is 3. \begin{align*}38\div 4\end{align*} is 9 with two left over. So, the 36th number is 7, the 37th number is 1, and the 38th number is 3.

b. The 72nd number is 7. \begin{align*}72 \div 4 \end{align*} is 18 with none left over. So, the 72nd number is 7.
c. The sum of the first 72 numbers is 288. The sum of one set of 1, 3, 5, 7 is 16. From the answer to question b, we know that 18 sets of four numbers ends with 72. The sum of the first 72 numbers is \begin{align*}16 \times 18=288\end{align*}.

### Practice

Look at each pattern and answer the questions.

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Date Created:
Jan 18, 2013